8.Probability.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "EE 502 P: Analytical Methods for Electrical Engineering\n",
    "# 8. Probability\n",
    "## Prof. Eric Klavins, Electrical and Computer Engineering\n",
    "### 25 September, 2019\n",
    "Copyright &copy; 2019, University of Washington"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-18T20:37:19.382579Z",
     "start_time": "2019-11-18T20:37:17.698445Z"
    },
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import networkx as nx\n",
    "import math\n",
    "import scipy\n",
    "import scipy.integrate as spi\n",
    "import numpy as np\n",
    "import sympy as sm\n",
    "sm.init_printing(use_latex='mathjax')\n",
    "import matplotlib.pyplot as plt\n",
    "import itertools\n",
    "import random\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Basics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Definitions\n",
    "\n",
    "**Def:** The **sample space**, $\\Omega$, of an experiment is the set of outcomes of the experiment. Subsets of $\\Omega$ are called **events**.\n",
    "\n",
    "**Example:** Consider the roll of two fair dice. The outcomes are the set of pairs in. \n",
    "\n",
    "$$\n",
    "\\Omega = \\{ 1,2,3,4,5,6 \\} \\times \\{ 1,2,3,4,5,6 \\}.\n",
    "$$\n",
    "\n",
    "An example event is the set of pairs in which the sum of the dice equals 7:\n",
    "\n",
    "$$\n",
    "B = \\{ (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \\} .\n",
    "$$\n",
    "\n",
    "**Def**: A **probability distribution** is a function $P : \\mathcal{P}(\\Omega) \\rightarrow [0,1]$ such that\n",
    "\n",
    "a) $P[A] \\geq 0$\n",
    "\n",
    "b) $P[\\Omega] = 1$\n",
    "\n",
    "c) $P[A \\cup B] = P[A] + P[B]$ whenever $A \\cap B = \\emptyset$. \n",
    "\n",
    "**Example:** The assignment $P[A] = \\frac{1}{36}|A|$ is a probability distribution for the dice example above. Property (a) is satisfied because the absolute value is positive. Property (b) is satisfied because there are 36 elements in $\\Omega$. Property (c) is satisfied because if $A$ and $B$ are disjoint sets then $|A+B| = |A| + |B|$. \n",
    "\n",
    "**Example:** The probability of the event $B$ above (the dice sum to seven) is $P[B] = \\frac{6}{36} = \\frac{1}{6}$. \n",
    "\n",
    "**Other Properties:**\n",
    "\n",
    "d) $P[\\emptyset] = 0$. This is because $P[\\Omega \\cup \\emptyset] = P[\\Omega] + P[\\emptyset] = 1$ by property (c). Since, by property (b), $P[\\Omega] = 1$, it must be the case that $P[\\emptyset] = 0$. \n",
    "\n",
    "e) $P[A \\cap \\overline B] = P[A] - P[A \\cap B]$. This is because $A \\cap B$ and $A \\cap \\overline{B}$ are disjoint. \n",
    "\n",
    "f) $P[A] = 1 - P[\\overline{A}]$. Take $A = \\Omega$ in (e). \n",
    "\n",
    "g) $P[A \\cup B] = P[A] + P[B] - P[A \\cap B]$ when $A \\cap B \\neq \\emptyset$. This can be shown using (e) along with the fact that $A \\cup B$ is equal to \n",
    "\n",
    "$$\n",
    "(A \\cap \\overline B) \\cup (A \\cap B) \\cup (\\overline{A} \\cap B).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Conditional Probability\n",
    "\n",
    "**Def:** The **probability of $B$ given $A$** is \n",
    "\n",
    "$$\n",
    "P[B\\;|\\;A] = \\frac{P[A\\cap B]}{P[A]}. \n",
    "$$\n",
    "\n",
    "It is the probability of the event $B$ given that $A$ has occurred.\n",
    "\n",
    "**Example:** With our running example of two dice being thrown, suppose that $A$ is the event that the face shown by the first die is even and $B$ is the event that the sum of the two dice is 7. Then\n",
    "\n",
    "$$\n",
    "P[A \\cap B] = P[ \\{ (2,5), (4,3), (6,1) \\} ] = \\frac{3}{36} = \\frac{1}{12}\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "P[A] = P[\\{ 2,4,6 \\} \\times \\{ 1,2,3,4,5,6 \\}]  = \\frac{18}{36} = \\frac{1}{2}.\n",
    "$$\n",
    "\n",
    "Thus, \n",
    "\n",
    "$$\n",
    "P[B\\;|\\;A] = \\frac{\\frac{1}{12}}{\\frac{1}{2}} = \\frac{1}{6}.\n",
    "$$\n",
    "\n",
    "That is, given that the first die comes up even, there is (still) a one in six chance that the sum of the two dice turns out to be seven.\n",
    "\n",
    "**Example:** The probability that the sum is equal to $a$ given that it is equal to $b$ is 1 if $a=b$ and 0 otherwise.\n",
    "\n",
    "**Property:** Suppose that $A_1$, .., $A_n$ are $n$ mutually exclusive events whose union is $\\Omega$. Then\n",
    "\n",
    "$$\n",
    "P[B] = P[B\\;|\\;A_1] + \\dots  + P[B\\;|\\;A_n] .\n",
    "$$\n",
    "\n",
    "**Example:** Suppose $A_i = \\{i\\} \\times \\{1,2,3,4,5,6\\}$ for $i = 1$ to $6$. Then if $B$ is the probability of getting a seven, we have\n",
    "\n",
    "\\begin{eqnarray}\n",
    "P[B] & = & P[B\\;|\\;A_n] + \\dots + P[B\\;|\\;A_n] \\\\\n",
    "     & = & \\frac{1}{36} + \\dots + \\frac{1}{36} \\\\\n",
    "     & = & \\frac{1}{6}.\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##  Independence\n",
    "\n",
    "**Def:** Two events $A$ and $B$ are **independent** if $P[A \\cap B] = P[A]P[B]$. \n",
    "\n",
    "**Property:** If $A$ and $B$ are independent, then $P[B \\;|\\; A] = P[B]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Bayes Theorem\n",
    "\n",
    "**Theorem:** Let $A$ and $B$ be events. Then\n",
    "    \n",
    "$$\n",
    "P[A \\;|\\; B] = \\frac{P[B\\;|\\;A]P[A]}{P[B]}.\n",
    "$$\n",
    "\n",
    "**Example:** (Stark and Woods p. 21) Suppose there is a test for cancer with the following properties\n",
    "\n",
    "- $A$ = the event that the test says the person has cancer\n",
    "- $B$ = the event that the person actually has cancer\n",
    "\n",
    "Suppose that \n",
    "\n",
    "- $P[A \\;|\\; B] = 0.95$: Test returns true if patient has cancer.\n",
    "- $P[\\overline{A} \\;|\\; \\overline{B}] = 0.95$ Test returns false if patient does not have cancer.\n",
    "- $P[B] = 0.005$.\n",
    "\n",
    "Is this a good test?\n",
    "\n",
    "To answer this question, we look at the probability that the person has cancer given the test comes up positive. That is,\n",
    "\n",
    "\\begin{eqnarray}\n",
    "P[B \\;|\\; A] & = & \\frac{P[A\\;|\\;B]P[B]}{P[A]} \\\\\n",
    "             & = & \\frac{P[A\\;|\\;B]P[B]}{P[A\\;|\\;B]P[B] + P[A\\;|\\;\\overline{B}]P[\\overline{B}]}\n",
    "\\end{eqnarray}\n",
    "\n",
    "where we have expanded $P[A]$ using the conditional probability property above. To compute $P[A \\;|\\; \\overline{B}]$ note that\n",
    "\n",
    "$$\n",
    "P[\\overline{A} \\;|\\; \\overline{B}] = \n",
    "\\frac{P[\\overline{A} \\cap \\overline{B}]}{P[\\overline{B}]} = \\frac{P[\\overline{B}] - P[A \\cap \\overline{B}]}{P[\\overline{B}]} = 1 - P[A\\;|\\;\\overline{B}]\n",
    "$$\n",
    "\n",
    "using property (e). Thus, \n",
    "\n",
    "\\begin{eqnarray}\n",
    "P[B \\;|\\; A] & = & \\frac{0.95 \\cdot 0.005}{0.95 \\cdot 0.005 + 0.05 \\cdot 0.995 } \\\\\n",
    "             & = & 0.087.\n",
    "\\end{eqnarray}\n",
    "\n",
    "The probability the patient has cancer given that the test was positive is only 8.7%, so this is not a very good test."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-19T19:27:48.816074Z",
     "start_time": "2019-11-19T19:27:48.811554Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Random Variables\n",
    "\n",
    "**Def:** A **random variable** $X$ is a function $X : \\Omega \\rightarrow \\mathbb{R}$ such that \n",
    "\n",
    "i) Given a measureable set $B \\subseteq \\mathbb{R}$, the set\n",
    "\n",
    "$$\\{ \\omega \\in \\Omega \\;|\\; X(\\omega) \\in B \\}$$\n",
    "\n",
    "is an event (this is trivially satisfied if $\\Omega$ is a discrete set).\n",
    "\n",
    "ii) $\\lim_{x \\rightarrow \\infty}P[X=x] = \\lim_{x \\rightarrow -\\infty}P[X=x] = 0$. \n",
    "\n",
    "This is a tricky definition and usually we don't worry about its details. Examples clarify.\n",
    "\n",
    "**Example:** Consider the two dice example and define $X$ to be the sum of the two dice. Then we find probabilities such as\n",
    "\n",
    "$$P[X = 7] = P[\\omega \\in \\Omega \\;|\\; X(\\omega) = 7] = 1/6.$$\n",
    "\n",
    "Similarly, \n",
    "\n",
    "$$\n",
    "P[X=2] = 1/36\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "P[X=13] = 0. \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cumulative Density Function\n",
    "\n",
    "**Def:** Given a random variable $X$, the **cumulative density function** (CDF) for $X$ is\n",
    "    \n",
    "$$\n",
    "F_X(x) = P[X \\leq x].\n",
    "$$\n",
    "\n",
    "**Example:** If $X$ is the sum of the two dice, the CDF determined by summing up the probabilities of getting a sum less than or equal to $x$, since all such events are disjoint. Thus, the CDF is as follows:\n",
    "\n",
    "\\begin{eqnarray}\n",
    "F_X(x) & = & 0 \\; \\mathrm{for} \\; x \\leq 1 \\\\\n",
    "F_X(2) & = & 1/36 \\\\\n",
    "F_X(3) & = & 3/36 \\\\\n",
    "F_X(4) & = & 6/36 \\\\\n",
    "F_X(5) & = & 10/36 \\\\\n",
    "F_X(6) & = & 15/36 \\\\\n",
    "F_X(7) & = & 21/36 \\\\\n",
    "F_X(8) & = & 26/36 \\\\\n",
    "F_X(9) & = & 30/36 \\\\\n",
    "F_X(10) & = & 33/36 \\\\\n",
    "F_X(11) & = & 35/36 \\\\\n",
    "F_X(12) & = & 36/36 \\\\\n",
    "F_X(x) & = & 36/36 \\; \\mathrm{for} \\; x \\leq 0 .\n",
    "\\end{eqnarray}\n",
    "\n",
    "## The Probability density function\n",
    "\n",
    "**Def:** If $\\Omega$ is discrete, then the **probability density function** (PDF) of a random variable $X$ is\n",
    "\n",
    "$$\n",
    "f_X(x) = P[\\{ \\omega \\;|\\; X(\\omega) = x\\}].\n",
    "$$\n",
    "\n",
    "**Example:** For the dice example where $X$ is the sum of the two dice,\n",
    "\n",
    "\\begin{eqnarray}\n",
    "f_X(x) & = & 0 \\; \\mathrm{for} \\; x \\leq 0 \\\\\n",
    "f_X(2) & = & 1/36 \\\\\n",
    "f_X(3) & = & 2/36 \\\\\n",
    "f_X(4) & = & 3/36 \\\\\n",
    "f_X(5) & = & 4/36 \\\\\n",
    "f_X(6) & = & 5/36 \\\\\n",
    "f_X(7) & = & 6/36 \\\\\n",
    "f_X(8) & = & 5/36 \\\\\n",
    "f_X(9) & = & 4/36 \\\\\n",
    "f_X(10) & = & 3/36 \\\\\n",
    "f_X(11) & = & 2/36 \\\\\n",
    "f_X(12) & = & 1/36 \\\\\n",
    "f_X(x) & = & 0/36 \\; \\mathrm{for} \\; x \\leq 0 .\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T00:00:54.689549Z",
     "start_time": "2019-11-20T00:00:54.491179Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAEICAYAAACktLTqAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0,14,15);\n",
    "CDF = np.array([0,0,1,3,6,10,15,21,26,30,33,35,36,36,36]) / 36;\n",
    "PDF = np.array([0,0,1,2,3,4,5,6,5,4,3,2,1,0,0]) / 36;\n",
    "plt.plot(x,CDF, \"o\", label=\"F(x)\")\n",
    "plt.plot(x,PDF, \"o\", label=\"f(x)\")\n",
    "plt.legend();\n",
    "plt.title(\"Sum of two dice\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Properties of the CDF:**\n",
    "\n",
    "a) $F(\\infty) = 1$ and $F(-\\infty) = 0$. \n",
    "\n",
    "b) $x \\leq y$ implies that $F(x) \\leq F(x)$. Thus, $F$ is a non-decreasing function.\n",
    "\n",
    "**Properties of the PDF:**\n",
    "\n",
    "a) $\\int_{-\\infty}^\\infty f_X(x) dx = 1$\n",
    "\n",
    "b) $F_X(x) = \\sum f_X(x)$, assuming $x$ takes on discrete values. \n",
    "\n",
    "**Def:** If $F_X$ is continuous and differentiable, then \n",
    "\n",
    "$$\n",
    "f_X(x) = \\frac{d}{dx}F_X(x) \n",
    "$$\n",
    "\n",
    "and the above sum in (b) should be an integral. \n",
    "\n",
    "**Example:** Suppose that $T$ represents the waiting time for the radioactive decay of an atom. This can be modeled by an exponential distribution with parameter $\\lambda$ with\n",
    "\n",
    "$$\n",
    "f_T(t) = \\lambda e^{-\\lambda t}\n",
    "$$\n",
    "\n",
    "when $t \\geq 0$ and $0$ otherwise.\n",
    "\n",
    "It can be checked that $f$ integrates to 1 over $[0,\\infty)$ and the CDF is defined by\n",
    "\n",
    "$$\n",
    "F_X(t) = \\int_0^\\infty \\lambda e^{-\\lambda t} dt = 1 - e^{-\\lambda t}. \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Note:** If $X$ is more than one dimensional, then the above definitions must be extended to $F(x_1, x_2, ... x_n)$. We will not cover that in these notes. The extension is straightforward, but requires a fair amount of book keeping."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Functions of random variables\n",
    "\n",
    "If $X$ is a random variable, we can make other random variables from $X$ such as $Y = X^2$ or $Z = 1/(1+X)$. Each of these new random variables also has a CDF and PDF. \n",
    "\n",
    "**Example:** Let $Y = X^2$ in the dice example. Then $Y$ is the square of the sum of the two dice. It has values $4$, $9$, $16$, $25$, $36$, and so on. For example, the probability that $Z$ is $36$ is the same as the probability that $X$ is $6$ which is $10/36$. \n",
    "\n",
    "**Example:** If $Y = T^2$ and the PDF of $T$ is exponentially distributed, then\n",
    "\n",
    "\\begin{eqnarray}\n",
    "F_U(u) & = & P[U \\leq u] = P[T^2 < u] \\\\\n",
    "       & = & P[-\\sqrt{u} \\leq T \\leq \\sqrt{u}] \\\\\n",
    "       & = & P[T \\leq \\sqrt{u}] \\\\\n",
    "       & = & F_T(\\sqrt{u}) \\\\\n",
    "       & = & 1 - e^{-\\lambda \\sqrt{u}}. \n",
    "\\end{eqnarray}\n",
    "\n",
    "The third equality above is due to the fact that the probability $T$ is negative is $0$ for this particular distribution. \n",
    "\n",
    "To compute the PDF, we have\n",
    "\n",
    "\\begin{eqnarray}\n",
    "f_U(u) & = & \\frac{d}{dt} \\left (1 - e^{-\\lambda \\sqrt{u}} \\right ) \\\\\n",
    "       & = & \\frac{1}{2 \\sqrt{u}} \\lambda e^{-\\lambda \\sqrt{u}} .\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T00:03:18.908244Z",
     "start_time": "2019-11-20T00:03:18.667896Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAEMCAYAAAA/Jfb8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.linspace(0,2,1000)\n",
    "u = np.linspace(0,2,1000)\n",
    "l = 1\n",
    "plt.plot(t,1-np.exp(-l*t), label=\"$F_T$\")\n",
    "plt.plot(t,1-np.exp(-l*np.sqrt(u)), label=\"$F_U$\")\n",
    "plt.title(\"CDFs for $T$ and $U = T^2$\")\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Moments, means, and variances\n",
    "\n",
    "**Def:** For a discrete system, the **expected value** of a random variable $X$ is \n",
    "\n",
    "$$\n",
    "E[X] = \\sum_x x \\;f_X(x).\n",
    "$$\n",
    "\n",
    "For a real valued system, the **expected value** of a random variable $X$, also called the **mean** $\\mu$ of $X$, is \n",
    "\n",
    "$$\n",
    "\\mu = E[X] = \\int_{\\infty}^\\infty x \\;f_X(x) dx.\n",
    "$$\n",
    "\n",
    "If $Y = g(X)$ is a function of the random variable $X$, then we can compute its expected value as well.\n",
    "\n",
    "$$\n",
    "E[Y] = E[g(X)] \\int_{-\\infty}^\\infty y \\;f_Y(x) dy. \n",
    "$$\n",
    "\n",
    "It turns out that to compute $E[Y]$ you don't need $f_Y$. You can use:\n",
    "\n",
    "$$\n",
    "E[Y] = \\int_{-\\infty}^\\infty g(x) \\;f_X(x) dx . \n",
    "$$\n",
    "\n",
    "**Example:** With the dice example and $X$ representing the sum of the two dice, we have\n",
    "\n",
    "$$\n",
    "E[X] = 2\\cdot\\frac{1}{36}\n",
    "     + 3\\cdot\\frac{2}{36}\n",
    "     + 4\\cdot\\frac{3}{36}\n",
    "     + 5\\cdot\\frac{4}{36}\n",
    "     + 6\\cdot\\frac{5}{36}\n",
    "     + 7\\cdot\\frac{6}{36}\n",
    "     + 8\\cdot\\frac{5}{36}\n",
    "     + 9\\cdot\\frac{4}{36}\n",
    "     + 10\\cdot\\frac{3}{36}\n",
    "     + 11\\cdot\\frac{2}{36}     \n",
    "     + 12\\cdot\\frac{1}{36} = 7.\n",
    "$$\n",
    "\n",
    "The expected value of the sum of the square is \n",
    "\n",
    "$$\n",
    "E[X^2] = 2^2\\cdot\\frac{1}{36}\n",
    "     + 3^2\\cdot\\frac{2}{36}\n",
    "     + 4^2\\cdot\\frac{3}{36}\n",
    "     + 5^2\\cdot\\frac{4}{36}\n",
    "     + 6^2\\cdot\\frac{5}{36}\n",
    "     + 7^2\\cdot\\frac{6}{36}\n",
    "     + 8^2\\cdot\\frac{5}{36}\n",
    "     + 9^2\\cdot\\frac{4}{36}\n",
    "     + 10^2\\cdot\\frac{3}{36}\n",
    "     + 11^2\\cdot\\frac{2}{36}     \n",
    "     + 12^2\\cdot\\frac{1}{36} = 54.83.\n",
    "$$\n",
    "\n",
    "**Example:** The expected value of the exponential distribution is\n",
    "\n",
    "$$\n",
    "E[T] = \\int_0^\\infty t \\lambda e^{-\\lambda t} dt = \\frac{1}{\\lambda}. \n",
    "$$\n",
    "\n",
    "Furthermore,\n",
    "\n",
    "$$\n",
    "E[T^2] = \\int_0^\\infty t \\lambda e^{-\\lambda t} dt = \\frac{1}{\\lambda} = \\frac{2}{\\lambda^2}. \n",
    "$$\n",
    "\n",
    "**Def:** The **nth moment** of $X$ is $E[X^n]$. The **nth** central moment of $X$ is $E[(X-\\mu)^n]$. The second central moment is also called the **variance** of $X$ and is denoted $\\sigma^2$. The **standard deviation** of $X$ is $\\sigma$. \n",
    "\n",
    "**Property:** \n",
    "\n",
    "$$\n",
    "\\sigma^2 = E[(X-\\mu)^2] = E[X^2] - E[2\\mu X] + E[\\mu^2].\n",
    "$$\n",
    "\n",
    "Since $E[2\\mu X] = 2\\mu E[X] = 2 \\mu^2$ we get\n",
    "\n",
    "$$\n",
    "\\sigma^2 = E[(X-\\mu)^2] = E[X^2] - \\mu^2.\n",
    "$$\n",
    "\n",
    "**Example:** The variance of $X$ in the dice problem is $\\sigma^2 = 54.83 - 49 = 5.83$. \n",
    "\n",
    "**Example:** The variance of the exponentially distributed $T$ is $\\frac{2}{\\lambda^2} - \\frac{1}{\\lambda^2} = \\frac{1}{\\lambda^2}$. The standard deviation is then $1 / \\lambda$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-19T23:26:42.916919Z",
     "start_time": "2019-11-19T23:26:42.880747Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\begin{cases} \\frac{1}{\\lambda} & \\text{for}\\: \\left|{\\operatorname{periodic_{argument}}{\\left (\\lambda,\\infty \\right )}}\\right| < \\frac{\\pi}{2} \\\\\\int_{0}^{\\infty} \\lambda t e^{- \\lambda t}\\, dt & \\text{otherwise} \\end{cases}, \\quad \\begin{cases} \\frac{2}{\\lambda^{2}} & \\text{for}\\: \\left|{\\operatorname{periodic_{argument}}{\\left (\\lambda,\\infty \\right )}}\\right| < \\frac{\\pi}{2} \\\\\\int_{0}^{\\infty} \\lambda t^{2} e^{- \\lambda t}\\, dt & \\text{otherwise} \\end{cases}\\right )$$"
      ],
      "text/plain": [
       "⎛⎧      1                                         π  ⎧      2                 \n",
       "⎜⎪      ─         for │periodic_argument(λ, ∞)│ < ─  ⎪      ──         for │pe\n",
       "⎜⎪      λ                                         2  ⎪       2                \n",
       "⎜⎪                                                   ⎪      λ                 \n",
       "⎜⎪∞                                                  ⎪                        \n",
       "⎜⎨⌠                                                , ⎨∞                       \n",
       "⎜⎪⎮      -λ⋅t                                        ⎪⌠                       \n",
       "⎜⎪⎮ λ⋅t⋅ℯ     dt              otherwise              ⎪⎮    2  -λ⋅t            \n",
       "⎜⎪⌡                                                  ⎪⎮ λ⋅t ⋅ℯ     dt         \n",
       "⎜⎪0                                                  ⎪⌡                       \n",
       "⎝⎩                                                   ⎩0                       \n",
       "\n",
       "                         π⎞\n",
       "riodic_argument(λ, ∞)│ < ─⎟\n",
       "                         2⎟\n",
       "                          ⎟\n",
       "                          ⎟\n",
       "                          ⎟\n",
       "                          ⎟\n",
       "                          ⎟\n",
       "     otherwise            ⎟\n",
       "                          ⎟\n",
       "                          ⎠"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l, t = sm.symbols(\"lambda,t\")\n",
    "sm.integrate(t * l * sm.exp(-l*t), (t,0,sm.oo)), sm.integrate(t**2 * l * sm.exp(-l*t), (t,0,sm.oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-18T17:09:54.399394Z",
     "start_time": "2019-11-18T17:09:54.360703Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Discrete Time, Discrete State Markov Processes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Discrete processes\n",
    "\n",
    "**Def:** A **discrete random process** is a sequence of random variables $X_0$, $X_1$, ... $X_n$, ... that may be infinitely or finitely long.\n",
    "\n",
    "**Def:** A **sample trajectory ** of a discrete random process is a sequence of samples for each of the random variables. \n",
    "\n",
    "**Example 1:** Roll a die over and over again and define $X_i$ to be the value of the roll number $i$. One possible trajectory is the sequence $1,1,1,1,1,\\dots$ where all the dice turn up $1$ (which would be quite suspicious). \n",
    "\n",
    "**Example 2:** Roll a die over and over and let $Y_i$ be the sum of the rolls so far. An example trajectory is $1,2,3,4,...$, indicated a 1 each roll.\n",
    "\n",
    "**Example 3:** Put three quarters, three dimes, three nickels, and three pennies in a jar. At each step, randomly remove a coin from the jar. Let $Z_i$ be the sum of the coins removed from the jar after $i$ steps. An example trajectory is:\n",
    "\n",
    "$$\n",
    "25, 50, 75, 80, 85, 90, 100, 110, 120, 121, 122, 123\n",
    "$$\n",
    "\n",
    "**Def:** The probability of a trajectory $(x_i)$ is \n",
    "\n",
    "$$\n",
    "P[X_1 = x_1, X_2 = x_2, ...].\n",
    "$$\n",
    "\n",
    "**Example:** when rolling a die over and over again, the rolls are independent. Thus, \n",
    "\n",
    "$$\n",
    "P[X_1 = x_1, X_2 = x_2, ...] = P[X_1-x_1]P[X_2=x_2] ... = \\prod_{i=0}^\\infty P[X_i=x_i].\n",
    "$$\n",
    "\n",
    "Note that since all these probabilities are less than one, the product above is zero for any particular infinite sequence of numbers. So the probability of getting a 1 forever is zero. In fact the probability of getting any particular sequence is zero.\n",
    "\n",
    "We can obtain the probability of getting 100 ones in a row and then getting anything after that as\n",
    "\n",
    "$$\n",
    "\\prod_{i=0}^{100} P[X_i=x_i] = 2.34 \\times ⋅10^{−156} .\n",
    "$$\n",
    "\n",
    "**Note:** The set of trajectories starting with a particular finite sequence $w$ is denoted $\\uparrow w$ and is called the **upset** of the sequence $w$. It can be shown that all measurable sets of sequences can be obtained from finite intersections and infinite unions of upsets. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## The Markov property\n",
    "\n",
    "**Def:** A process is **Markov** if \n",
    "\n",
    "$$\n",
    "P[X_{k+1} = x_{k+1} \\;|\\; X_k = x_k, ..., X_0 = x_0 ] = P[X_{k+1} = x_{k+1} \\;|\\; X_k = x_k ].\n",
    "$$\n",
    "\n",
    "That is, the probability that the next state is equal to some value is dependent only on the current state.\n",
    "\n",
    "**Example:** Example 1 above is Markov because all the rolls are independent. In fact, the probability that $X_{k+1}$ is equal to any particular value does not depend even on $X_k$. \n",
    "\n",
    "**Example:** Example 2 above is Markov. The sum of the rolls up to step $k+1$ depends on the sum at step $k$ and the roll at step $k+1$ only. \n",
    "\n",
    "**Example:** Example 3 above is not Markov. For example, suppose that the sum at step 2 is 30. This could have been obtained by drawing a quarter and a nickel, drawing three dimes, or two dimes and two nickels. In each of those cases, the probability that a given coin is going to be drawn in step three is different.\n",
    "    \n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Markov Processes as Graphs\n",
    "\n",
    "**Def:** A Discrete time, discrete state Markov Process (also commonly called a Markov Chain) can be specified using a directed graph whose edges are labeled by a transition probability. In particular, let $G=(V,E)$ be a *directed* graph and let $Q:E \\rightarrow [0,1]$ be such that for all $u \\in V$\n",
    "\n",
    "$$\n",
    "\\sum_{v \\in \\Gamma(v)} Q(u,v) = 1.\n",
    "$$\n",
    "\n",
    "With this specification, the state $X_i$ is a random variable whose values range over $V$ and\n",
    "\n",
    "$$\n",
    "P[X_{k+1} = v \\;|\\; X_k = u] = Q(u,v). \n",
    "$$\n",
    "\n",
    "**Example:** Consider the following somewhat unrealistic model of the weather. It represents the following:\n",
    "\n",
    "- $V = \\{ 0, 1 \\}$ representing sunny and rainy weather.\n",
    "- $E = V \\times V$ where $(u,v)$ represents the weather $v$ on day $2$ given the weather $u$ on day one.\n",
    "- $Q(0,0) = 0.9$, $Q(0,1) = 0.1$, $Q(1,0) = 0.5$, and $Q(1,1) = 0.5$. \n",
    "\n",
    "<img src=\"https://raw.githubusercontent.com/klavins/EE502P/master/images/mp1.jpeg\" width=60%>\n",
    "\n",
    "An example trajectory is\n",
    "\n",
    "$$\n",
    "0,0,0,0,1,1,1,1, ...\n",
    "$$\n",
    "\n",
    "Assuming all trajectories start with sun, the probability of trajectories with the same first eight states as the above is\n",
    "\n",
    "$$\n",
    "0.9^3 \\cdot 0.1 \\cdot 0.5^3 = 0.0015625. \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Properties of Markov Processes\n",
    "\n",
    "**Def:** A Markov Process is called **irreducible** if it is there is a path from any state to any other state. Otherwise it is called **reducible**. \n",
    "\n",
    "**Def:** A state $x$ has **period** $k$ if starting in state $x$, if $x$ is returned to, it is returned to in $mk$ steps for some $m$. If $k=1$ for a given state, then it is called **aperiodic**. \n",
    "\n",
    "**Def:** A state is **transient** if there is a nonzero probability that it will not be returned to once left. Otherwise the state is called **recurrent**. \n",
    "\n",
    "**Def:** A state is called **absorbing** if it cannot be left once reached. A subset of states is also called **absorbing** if it cannot be left once reached. \n",
    "\n",
    "**Def:** A state is **ergodic** if it is aperiodic and there is a nonzero probability that it will be visited an infinite number of times in any trajectory.\n",
    "\n",
    "**Example:** Consider the following Markov Processes:\n",
    "\n",
    "<img src=\"https://raw.githubusercontent.com/klavins/EE502P/master/images/Four-MPs.jpeg\">\n",
    "\n",
    "- In process (a): States 0 and 1 have period 2. State 2 is absorbing. The process is not irreducible. State 2 is ergodic. States 0 and 1 are transient.\n",
    "\n",
    "- In process (b), all states have period 2. The set $\\{2,3\\}$ is absorbing. No states are ergodic. The system is not irreducible. States 0 and 1 are transient.\n",
    "\n",
    "- In process (c), all states have period 3. The process is irreducible. No states are ergodic. All states are recurrent.\n",
    "\n",
    "- In process (d), all states are aperiodic and ergodic.All states are recurrent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Update Matrix\n",
    "\n",
    "**Def:** A **Stochastic Matrix** $Q$ is a real valued $n \\times n$ matrix such that\n",
    "\n",
    "- $Q_{i,j} \\in [0,1]$\n",
    "- The sum of each row is one. \n",
    "\n",
    "**Note:** The assignment $Q$ in a graph specification of a discrete time, discrete state Markov Process is a stochastic matrix. \n",
    "\n",
    "**Example:** The weather example above has \n",
    "\n",
    "$$\n",
    "Q = \\begin{pmatrix}\n",
    "0.9 & 0.1 \\\\\n",
    "0.5 & 0.5\n",
    "\\end{pmatrix} .\n",
    "$$\n",
    "\n",
    "**Def:** A **probability vector** $p \\in [0,1]^n$ is a <u>row vector</u> such that \n",
    "\n",
    "$$\n",
    "p\\mathbb{1} = \\mathbb{1}^T p^T = 1. \n",
    "$$\n",
    "\n",
    "where $\\mathbb{1}$ is the column vector of all ones.\n",
    "\n",
    "**Property:** Let the probability vector $p_i$ correspond to the PDF $f_{X_i}$ for a Markov Process. Then,\n",
    "\n",
    "$$\n",
    "p_{k+1} = p_k Q. \n",
    "$$\n",
    "\n",
    "**Example:** For the weather system, suppose that \n",
    "\n",
    "$$\n",
    "p_0 = (1 \\; 0)\n",
    "$$\n",
    "\n",
    "Then, \n",
    "\n",
    "$$\n",
    "p_1 = (1\\;0) \\begin{pmatrix}\n",
    "0.9 & 0.1 \\\\\n",
    "0.5 & 0.5\n",
    "\\end{pmatrix} = (0.9 \\;\\; 0.1)\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "p_2 = (0.9 \\;\\; 0.1)\n",
    "\\begin{pmatrix}\n",
    "0.9 & 0.1 \\\\\n",
    "0.5 & 0.5\n",
    "\\end{pmatrix} = (0.86 \\;\\; 0.14).\n",
    "$$\n",
    "\n",
    "**Example:** Example (b) above has \n",
    "\n",
    "$$\n",
    "Q = \\begin{pmatrix}\n",
    "0 & 1 & 0 & 0 \\\\\n",
    "0.9 & 0   & 0.1 & 0 \\\\\n",
    "0 & 0 & 0 & 1 \\\\\n",
    "0 & 0   & 1 & 0 \n",
    "\\end{pmatrix} .\n",
    "$$\n",
    "\n",
    "Below is a simulation of the probability of being in each state versus time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 355,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T04:17:02.472632Z",
     "start_time": "2019-11-21T04:17:01.473931Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAt4AAAFACAYAAACLEHvxAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<Figure size 864x360 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = np.array([1,0,0,0]); # Initial probability vector\n",
    " \n",
    "Q = np.array([           # Transition probability matrix\n",
    "    [0,1,0,0],\n",
    "    [0.9,0,0.1,0],\n",
    "    [0,0,0.5,0.5],\n",
    "    [0,0,0.5,0.5]\n",
    "]);\n",
    "\n",
    "n = 50;\n",
    "x = [p]\n",
    "\n",
    "for _ in range(n):\n",
    "    p = p.dot(Q)   # Here's the update!\n",
    "    x.append(p)\n",
    "\n",
    "t = range(n+1)\n",
    "x = np.array(x)\n",
    "\n",
    "fig,ax = plt.subplots(2,2,figsize=(12,5))\n",
    "plt.subplots_adjust(hspace=0.35)\n",
    "for i in range(4):\n",
    "    a = ax[int(i/2)][i%2]\n",
    "    a.bar(t,x[:,i])\n",
    "    a.set_ylabel(\"$P[X_k=%d]$\" % i)\n",
    "    a.set_xlabel(\"k\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Properties of Stochastic Matrices\n",
    "\n",
    "**Property:** $1$ is a eigenvalue with eigenvector $\\mathbb{1}^T$. This is because every row sums to one. That is if $r$ is a row of $Q$, then $\\mathbb{1}r = 1$. Said differently,\n",
    "$$\n",
    "Q \\mathbb{1} = \\mathbb{1}. \n",
    "$$\n",
    "\n",
    "Note that the left and right eigenvectors of a matrix are the same, even though the eigenvalues may be different. Thus, $Q$ also has at least one left eigenvector of 1. \n",
    "\n",
    "**Def:** A **stationary probability vector $p^*$** has the property that\n",
    "\n",
    "$$\n",
    "p^* Q = p^*.\n",
    "$$\n",
    "\n",
    "Thus, $p^*$ is a left eigenvector of $Q$ with left eigenvalue 1. \n",
    "\n",
    "**Example:** Consider the weather example. The left eigenvalues and eigenvectors of \n",
    "\n",
    "$$\n",
    "Q = \\begin{pmatrix}\n",
    "0.9 & 0.1 \\\\\n",
    "0.5 & 0.5\n",
    "\\end{pmatrix} \n",
    "$$\n",
    "\n",
    "are \n",
    "\n",
    "$$\n",
    "\\frac{2}{5} \\; \\mathrm{with} \\; (-1\\;\\;1)\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "1 \\; \\mathrm{with} \\; (5\\;\\;1).\n",
    "$$\n",
    "\n",
    "This second eigenvector corresponds to the stationary distribution, although it does not sum to one. However, since every non-zero multiple of an eigenvector is also an eigenvector, we get:\n",
    "\n",
    "$$\n",
    "p^* = (\\frac{5}{6}\\;\\;\\frac{1}{6}).\n",
    "$$\n",
    "\n",
    "So over the years, with this model, it is five times more likely to be sunny than rainy. Must not be Seattle!\n",
    "\n",
    "**Example:** Consider the system (b) above with \n",
    "\n",
    "$$\n",
    "Q = \\begin{pmatrix}\n",
    "0 & 1 & 0 & 0 \\\\\n",
    "0.9 & 0   & 0.1 & 0 \\\\\n",
    "0 & 0 & 0 & 1 \\\\\n",
    "0 & 0   & 1 & 0 \n",
    "\\end{pmatrix} .\n",
    "$$\n",
    "\n",
    "It can be shown that a left eigenvector corresponding to $1$ is \n",
    "\n",
    "$$\n",
    "p^* = (0 \\; 0 \\; \\frac{1}{2} \\; \\frac{1}{2}).\n",
    "$$\n",
    "\n",
    "This makes sense, but it tricky to interpret. It is true the system is in state $2$ half the time at steady state, but only during even numbered time steps, which doesn't even really make sense at time equals infinity. \n",
    "\n",
    "**Example:** Suppose we have the system\n",
    "\n",
    "<img src=\"https://raw.githubusercontent.com/klavins/EE502P/master/images/2D-stationary.jpeg\" width=60%>\n",
    "\n",
    "which has \n",
    "\n",
    "$$\n",
    "Q = \\begin{pmatrix}\n",
    "0 & 0.4 & 0.6 \\\\\n",
    "0 & 1   & 0 \\\\\n",
    "0 & 0 & 1 \n",
    "\\end{pmatrix} .\n",
    "$$\n",
    "\n",
    "The left eigenvalues of this system are $0$, $1$, and $1$. The $1$ eigenvalues of left eigenvectors $(0,1,0)$ and $(0,0,1)$. But can a process have two separate distributions?\n",
    "\n",
    "No. Recall that if $x$ and $y$ are eigenvectors associated with the same eigenvalue, then so are $x+y$. Thus, the system has a two dimensional stationary state where all states have the form\n",
    "\n",
    "$$\n",
    "p^* = (0\\;\\; a \\;\\;\\; 1-a)\n",
    "$$\n",
    "\n",
    "with $a \\in [0,1]$. On the other hand it seems quite clear that the stationary state ought to be $(0\\;\\;0.4\\;\\;0.6)$. What is happening? In fact, the stationary analysis we did is valid, If you start with any probability vector of the form $(0\\;\\; a \\;\\;\\; 1-a)$, then you will stay with that distribution as the system progresses, so they are all stationary. \n",
    "\n",
    "However, only one of the stationary vectors is *reachable* from the initial distribution $p = (1\\;0\\;0)$. To determine this vector, we need to use the update rule:\n",
    "\n",
    "$$\n",
    "p_{k+1} = p_k Q .\n",
    "$$\n",
    "\n",
    "Note that by substituting in $p_{k-1}Q$ and iterating, we can show that\n",
    "\n",
    "$$\n",
    "p_{k} = p_0 Q^k\n",
    "$$\n",
    "\n",
    "If we find $Q_k$ and take the limit as $k \\rightarrow \\infty$ we should find the steady state distribution. In this case, $Q^k = Q$, so $Q^\\infty = Q$ (pretty unusual, so don't get used to this). Thus, \n",
    "\n",
    "$$\n",
    "p^* = (1 \\; 0 \\; 0 ) Q = (0\\;\\;0.4\\;\\;0.6).\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 343,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T01:41:00.779154Z",
     "start_time": "2019-11-21T01:41:00.767759Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}-1 & 5\\\\1 & 1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}\\frac{2}{5} & 0\\\\0 & 1\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "⎛⎡-1  5⎤, ⎡2/5  0⎤⎞\n",
       "⎜⎢     ⎥  ⎢      ⎥⎟\n",
       "⎝⎣1   1⎦  ⎣ 0   1⎦⎠"
      ]
     },
     "execution_count": 343,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Eigensystem of the sun/rain system\n",
    "\n",
    "Q = sm.Matrix([\n",
    "    [sm.Rational(9,10), sm.Rational(1,10)],\n",
    "    [sm.Rational(5,10), sm.Rational(5,10)]\n",
    "])\n",
    "\n",
    "(Q.T).diagonalize()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 344,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T01:41:01.799130Z",
     "start_time": "2019-11-21T01:41:01.761075Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}0 & 0 & 3 \\sqrt{10} & - 3 \\sqrt{10}\\\\0 & 0 & -10 & -10\\\\-1 & 1 & - 3 \\sqrt{10} & 3 \\sqrt{10}\\\\1 & 1 & 10 & 10\\end{matrix}\\right], \\quad \\left[\\begin{matrix}-1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & - \\frac{3 \\sqrt{10}}{10} & 0\\\\0 & 0 & 0 & \\frac{3 \\sqrt{10}}{10}\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "⎛⎡0   0  3⋅√10   -3⋅√10⎤, ⎡-1  0     0       0  ⎤⎞\n",
       "⎜⎢                     ⎥  ⎢                     ⎥⎟\n",
       "⎜⎢0   0   -10     -10  ⎥  ⎢0   1     0       0  ⎥⎟\n",
       "⎜⎢                     ⎥  ⎢                     ⎥⎟\n",
       "⎜⎢-1  1  -3⋅√10  3⋅√10 ⎥  ⎢       -3⋅√10        ⎥⎟\n",
       "⎜⎢                     ⎥  ⎢0   0  ───────    0  ⎥⎟\n",
       "⎜⎣1   1    10      10  ⎦  ⎢          10         ⎥⎟\n",
       "⎜                         ⎢                     ⎥⎟\n",
       "⎜                         ⎢                3⋅√10⎥⎟\n",
       "⎜                         ⎢0   0     0     ─────⎥⎟\n",
       "⎝                         ⎣                  10 ⎦⎠"
      ]
     },
     "execution_count": 344,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Eigensystem of process (b) above\n",
    "\n",
    "Q = sm.Matrix([\n",
    "    [0,1,0,0],\n",
    "    [sm.Rational(9,10), 0, sm.Rational(1,10), 0],\n",
    "    [0,0,0,1],\n",
    "    [0,0,1,0]\n",
    "])\n",
    "\n",
    "(Q.T).diagonalize()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 345,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T01:41:05.146789Z",
     "start_time": "2019-11-21T01:41:05.131922Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}-5 & 0 & 0\\\\2 & 1 & 0\\\\3 & 0 & 1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "⎛⎡-5  0  0⎤, ⎡0  0  0⎤⎞\n",
       "⎜⎢        ⎥  ⎢       ⎥⎟\n",
       "⎜⎢2   1  0⎥  ⎢0  1  0⎥⎟\n",
       "⎜⎢        ⎥  ⎢       ⎥⎟\n",
       "⎝⎣3   0  1⎦  ⎣0  0  1⎦⎠"
      ]
     },
     "execution_count": 345,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Eignensystem with a 2D stationary set\n",
    "\n",
    "Q = sm.Matrix([\n",
    "    [0,sm.Rational(4,10),sm.Rational(6,10)],\n",
    "    [0,1,0],\n",
    "    [0,0,1]\n",
    "])\n",
    "\n",
    "(Q.T).diagonalize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Example: A Birth Death Process\n",
    "\n",
    "Consider the infinite Markov process\n",
    "\n",
    "<img src=\"https://raw.githubusercontent.com/klavins/EE502P/master/images/birth-death-process\n",
    ".jpeg\" width=80%>\n",
    "\n",
    "This process could describe a queuing model where at each step: with probability $p$ a new customer arrives, with probability $q$ a customer is serviced, and with the remaining probability seemingly nothing happens.\n",
    "\n",
    "We wish to determine the mean number of customers at steady state, as well as the standard deviation of the number of customers. \n",
    "\n",
    "Abusing notation, let $p_i = p^*(i)$. Then at steady state\n",
    "\n",
    "\\begin{eqnarray}\n",
    "p_0 & = & (1-\\alpha) p_0 + \\beta p_1 \\\\\n",
    "p_1 & = & \\alpha p_0 + \\gamma p_1 + \\beta p_2 \\\\\n",
    "p_2 & = & \\alpha p_1 + \\gamma p_2 + \\beta p_3 \\\\\n",
    "    & \\vdots &\n",
    "\\end{eqnarray}\n",
    "\n",
    "where $\\gamma = 1 - \\alpha - \\beta$. Solving the first equation for $p_1$ gives\n",
    "\n",
    "$$\n",
    "p_1 = \\frac{\\alpha}{\\beta} p_0.\n",
    "$$\n",
    "\n",
    "Solving the second equation for $p_2$ and substituting in the above value for $p_1$ gives\n",
    "\n",
    "$$\n",
    "p_2 = \\frac{\\alpha^2}{\\beta^2} p_0.\n",
    "$$\n",
    "\n",
    "In general, $p_k = \\frac{\\alpha^k}{\\beta^k} p_0$. \n",
    "\n",
    "Now we use the fact that $\\sum_{k=0}^\\infty p_k = 1$ to get\n",
    "\n",
    "$$\n",
    "p_0 + \\frac{\\alpha}{\\beta} p_0 + \\frac{\\alpha^2}{\\beta^2} p_0 + \\cdots = p_0 \\sum_{k=0}^\\infty \\left ( \\frac{\\alpha}{\\beta} \\right ) ^ k = p_0 \\frac{1}{1 - \\frac{\\alpha}{\\beta}} = 1 .\n",
    "$$\n",
    "\n",
    "Thus, $p_0 = 1 - \\frac{\\alpha}{\\beta}$. In order for this to be a valid probability, we clearly need that $\\alpha \\leq \\beta$. Otherwise, the queue grows forever and there is no stationary distribution.\n",
    "\n",
    "Now, the mean at steady state is\n",
    "\n",
    "$$\n",
    "\\mu^* = \\left (1 - \\frac{\\alpha}{\\beta} \\right ) \\sum_{k=0}^\\infty k \\left ( \\frac{\\alpha}{\\beta} \\right )^k  = \\left (1 - \\frac{\\alpha}{\\beta} \\right ) \\frac{\\alpha}{\\beta(1-\\frac{\\alpha}{\\beta})^2} = \\frac{\\alpha}{\\beta - \\alpha}. \n",
    "$$\n",
    "\n",
    "We can also find the second moment, which is\n",
    "\n",
    "$$\n",
    "\\mu_2^* = \\left (1 - \\frac{\\alpha}{\\beta} \\right ) \\sum_{k=0}^\\infty k^2 \\left ( \\frac{\\alpha}{\\beta} \\right )^k = \\frac{\\alpha(\\alpha+\\beta)}{(\\alpha-\\beta)^2} .\n",
    "$$\n",
    "\n",
    "The standard deviation is then\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\sigma & = & \\sqrt{\\mu_2^* - (\\mu^*)^2 } \\\\\n",
    " & = & \\sqrt{\n",
    " \\frac{\\alpha(\\alpha+\\beta)}{(\\alpha-\\beta)^2} - \n",
    "\\left ( \\frac{\\alpha}{\\beta - \\alpha} \\right )^2\n",
    "} \\\\\n",
    "& = & \\frac{1}{\\beta-\\alpha}\\sqrt{\\alpha\\beta} .\n",
    "\\end{eqnarray}\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 561,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-22T00:12:42.229515Z",
     "start_time": "2019-11-22T00:12:42.067027Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left \\{ p_{1} : \\frac{a p_{0}}{b}, \\quad p_{2} : \\frac{a^{2} p_{0}}{b^{2}}, \\quad p_{3} : \\frac{a^{3} p_{0}}{b^{3}}, \\quad p_{4} : \\frac{a^{4} p_{0}}{b^{4}}\\right \\}$$"
      ],
      "text/plain": [
       "⎧               2          3          4   ⎫\n",
       "⎪    a⋅p₀      a ⋅p₀      a ⋅p₀      a ⋅p₀⎪\n",
       "⎨p₁: ────, p₂: ─────, p₃: ─────, p₄: ─────⎬\n",
       "⎪     b           2          3          4 ⎪\n",
       "⎩                b          b          b  ⎭"
      ]
     },
     "execution_count": 561,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solving the first four equations in terms of p0\n",
    "\n",
    "a,b,g,p0,p1,p2,p3,p4 = sm.symbols(\"a b g p0 p1 p2 p3 p4\")\n",
    "sol = sm.solve([\n",
    "    -p0 + (1-a)*p0 + b*p1,\n",
    "    -p1 + a*p0 + (1-a-b)*p1 + b*p2,\n",
    "    -p2 + a*p1 + (1-a-b)*p2 + b*p3,\n",
    "    -p3 + a*p2 + (1-a-b)*p3 + b*p4\n",
    "], [p1, p2, p3, p4] )\n",
    "sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 411,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T20:22:27.092664Z",
     "start_time": "2019-11-21T20:22:26.987376Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$- \\frac{a}{a - b}$$"
      ],
      "text/plain": [
       " -a  \n",
       "─────\n",
       "a - b"
      ]
     },
     "execution_count": 411,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solving for the mean\n",
    "\n",
    "a, b, k = sm.symbols(\"a b k\")\n",
    "mu = sm.Sum(k*(a/b)**k, (k,0,sm.oo)).doit().args[0][0]*(1-a/b)\n",
    "mu = sm.simplify(mu)\n",
    "mu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 412,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T20:22:31.950420Z",
     "start_time": "2019-11-21T20:22:31.812849Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{a \\left(a + b\\right)}{\\left(a - b\\right)^{2}}$$"
      ],
      "text/plain": [
       "a⋅(a + b)\n",
       "─────────\n",
       "        2\n",
       " (a - b) "
      ]
     },
     "execution_count": 412,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solving for the second moment\n",
    "\n",
    "mu2 = sm.Sum(k**2 * (a/b)**k, (k,0,sm.oo)).doit()\n",
    "mu2 = mu2.args[0][0]\n",
    "mu2 = sm.simplify((1-a/b)*mu2)\n",
    "mu2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 416,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T20:22:58.566353Z",
     "start_time": "2019-11-21T20:22:58.516993Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\sqrt{\\frac{a b}{\\left(a - b\\right)^{2}}}$$"
      ],
      "text/plain": [
       "     __________\n",
       "    ╱   a⋅b    \n",
       "   ╱  ──────── \n",
       "  ╱          2 \n",
       "╲╱    (a - b)  "
      ]
     },
     "execution_count": 416,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.simplify(sm.sqrt(mu2 - mu**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 354,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T01:44:12.433245Z",
     "start_time": "2019-11-21T01:44:11.047829Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAEKCAYAAAAfGVI8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4zLCBo\ndHRwOi8vbWF0cGxvdGxpYi5vcmcvnQurowAAIABJREFUeJztvXl8G9W5//8+smzLSpw4dvatiTFZ\nbCeExFkJkBsISctWWsrSQoELpZTSW0q3wG174ZbbC/dHod/2tqVwQ4FSEspOoaVA2VoogZAGSJwV\nJyEhexw7dmzZknV+f0hnGI212pZkW8/79fLL0mjmnGfOjObRzHzm8yitNYIgCELu4sp2AIIgCEJ2\nkUQgCIKQ40giEARByHEkEQiCIOQ4kggEQRByHEkEgiAIOY4kAkEQhBxHEoEgCEKOI4lAEAQhx3Gn\nq2GllAd4HSgM9/OY1vo/lFITgVVAGfAucKnWuj1eW0OHDtUTJkxIV6iCIAj9knffffeQ1npYovnS\nlgiANmCx1rpZKZUP/F0p9WfgBuAurfUqpdTdwJXAr+M1NGHCBNasWZPGUAVBEPofSqmdycyXtktD\nOkRz+G1++E8Di4HHwtMfAD6brhgEQRCExKT1HoFSKk8ptQ44ALwIfAg0aK0D4Vl2A2PSGYMgCIIQ\nn7QmAq11h9Z6BjAWmANMSXZZpdTVSqk1Sqk1Bw8eTFuMgiAIuU467xFYaK0blFKvAPOBEqWUO3xW\nMBb4OMYy9wD3ANTU1HTyyvb7/ezevRufz5fGyAUhPh6Ph7Fjx5Kfn5/tUAShy6RTNTQM8IeTQBGw\nBLgdeAU4n5By6DLg6a60v3v3boqLi5kwYQJKqZ4KWxCSRmvN4cOH2b17NxMnTsx2OILQZdJ5aWgU\n8IpS6n3gHeBFrfWzwPeBG5RS2whJSFd0pXGfz0dZWZkkASFrKKUoKyuTs1Khz5O2MwKt9fvAiVGm\n1xG6X9BtJAkI2Ub2QaE/kJF7BIIgCN1l+/btHDt2DK/XSzAYxOWKvKBhprW0tAAwevRoSktLsxFq\nn0MsJrpBXl4eM2bMsP5uu+22jPb/ox/9iJdeeinl5Xbs2EF1dXXMz3/2s5/h8XhobGzsTniC0KMc\nO3YMgJaWFnw+X8QlOfPePq2hoSHjMfZV5IygGxQVFbFu3bqs9N3R0cF//ud/pqXtlStXMnv2bJ54\n4gmuuOKKbrcXCARwu2VXE3oWj8dDeXk5AHV1dbS0tODxeKwzAiF55IwgDUyYMIEbb7yRGTNmUFNT\nw9q1a1m6dCnHHXccd999NwCvvvoqp5xyCmeeeSaTJ0/mmmuuIRgMAvDCCy8wf/58Zs6cyRe+8AWa\nm5utdr///e8zc+ZMHn30US6//HIeeyz0kPY777zDggULOOGEE5gzZw5NTU3s2LGDk08+mZkzZzJz\n5kzefPPNhLF/+OGHNDc3c+utt7Jy5Upr+rx589iwYYP1ftGiRaxZs4Zjx47xr//6r8yZM4cTTzyR\np58OicDuv/9+zjnnHBYvXsxpp51Gc3Mzp512GjNnzmTatGnWfAA//vGPmTx5MgsXLuTiiy/mjjvu\nsGJZtmwZs2bN4uSTT2bTpk3d2SyCIMSgX/xMu/566Okf5jNmwM9+Fn+e1tZWZsyYYb2/8cYbufDC\nCwEYP34869at41vf+haXX345b7zxBj6fj+rqaq655hoA3n77bWpra/nUpz7FsmXLeOKJJ1i0aBG3\n3norL730EgMGDOD222/nzjvv5Ec/+hEAZWVlrF27FoDnn38egPb2di688EIeeeQRZs+ezdGjRykq\nKmL48OG8+OKLeDwetm7dysUXX5zQs2nVqlVcdNFFnHzyyWzevJn9+/czYsQILrzwQv7whz9wyy23\nsHfvXvbu3UtNTQ033XQTixcv5r777qOhoYE5c+Zw+umnA7B27Vref/99SktLCQQCPPnkkwwaNIhD\nhw4xb948zjnnHNasWcPjjz/Oe++9h9/vZ+bMmcyaNQuAq6++mrvvvpvjjz+e1atXc+211/Lyyy+n\nuCUFQUhEv0gE2SLepaFzzjkHgGnTptHc3ExxcTHFxcUUFhZa1y7nzJljndpefPHF/P3vf8fj8VBb\nW8tJJ50EhA7y8+fPt9o1icbO5s2bGTVqFLNnzwZg0KBBQOia6nXXXce6devIy8tjy5YtCddp5cqV\nPPnkk7hcLj7/+c/z6KOPct1113HBBRdwxhlncMstt/CHP/yB888/HwidvTzzzDPWr3ifz8dHH30E\nwJIlS6ybdVprbrrpJl5//XVcLhcff/wx+/fv54033uDcc8/F4/Hg8Xg4++yzAWhububNN9/kC1/4\nghVbW1tbwvgFQUidfpEIEv1yzwaFhYUAuFwu67V5HwiErJac0kOlFFprlixZEnFZxs6AAQOSjuGu\nu+5ixIgRvPfeewSDQTweT9z5P/jgA7Zu3cqSJUuAUBKaOHEi1113HWPGjKGsrIz333+fRx55xLrE\npbXm8ccfZ/LkyRFtrV69OiLW3//+9xw8eJB3332X/Px8JkyYEFd/HwwGKSkpydo9GEHIJeQeQRZ5\n++232b59O8FgkEceeYSFCxcyb9483njjDbZt2waEftUn+iU/efJk9u7dyzvvvANAU1MTgUCAxsZG\nRo0ahcvl4ne/+x0dHR1x21m5ciU333wzO3bsYMeOHezZs4c9e/awc2fIyfbCCy/kf/7nf2hsbGT6\n9OkALF26lF/84hdoHXIB+ec//xm17cbGRoYPH05+fj6vvPKK1eZJJ53EH//4R3w+H83NzTz77LNA\n6Kxm4sSJPProo0Ao4bz33nsJx1TondTX11NXV0d9fX3U98ksnyo+ny9mH/b+t2/fzvbt21NuPxm2\nb9/O+vXrqaurY9u2bVHjyVQs8ZBE0A3MPQLzt3z58pSWnz17Ntdddx1Tp05l4sSJnHfeeQwbNoz7\n77+fiy++mOnTpzN//vyEN0kLCgp45JFH+MY3vsEJJ5zAkiVL8Pl8XHvttTzwwAOccMIJbNq0KeHZ\nxKpVqzjvvPMipp133nmsWrUKgPPPP59Vq1ZxwQUXWJ//8Ic/xO/3M336dKqqqvjhD38Yte0vfelL\nrFmzhmnTpvHggw8yZcoUawzOOeccpk+fzqc//WmmTZvG4MGDgdBZxIoVKzjhhBOoqqqKuMEs9C0a\nGhpoaWmxLos63yezvMHj8eD1eikpKbGmlZSUWNPMfu7xePD5fFH7sPd/7NgxS5ra0zglr9HWOVOx\nxEOZX3K9mZqaGu28yblx40amTp2apYi6z6uvvsodd9xh/QLOZZqbmxk4cCAtLS2ccsop3HPPPcyc\nOTPbYSVNX98XM4GRd3q9XsrLyzu9T2Z5IKl5k1nO3r+Rm8Z7tqarrF+/vtM05zqnMxal1Lta65pE\n8/WLewRC3+bqq6+mtrYWn8/HZZdd1qeSgCD0ByQRZIlFixaxaNGibIfRK3j44YezHYIg5DRyj0AQ\nBCHHkUQgCIKQ48ilIUEQepT6+noaGhosVU9DQ4P1zEhLSwvbtm3r9N7lclFSUtLJLdS05fP5Ej4H\nE4uWlhbq6+spLS2NaA+IeJalrq4uagxdJZbk1efzWZLwoUOHRkxPVyyJkDMCQRB6FLsc0n4QNwdy\nc8Czv48lJbUvb5eLJos9GTnbs8dk4uhJx9JoklfTZ3t7O+3t7Z3mSVcsiZBE0EUOHz5sPT8wcuRI\nxowZY71vb29Pa9/19fXWk71d5b777mPfvn3W+yuuuILNmzd3NzRB6IRxCa2oqMDr9QIhCaX9fTLL\nd+XXcWlpaac+7PGUl5dTXV1NdXV1l8844uH1eqmurrb6Mv+dtRScktJ0xBIPSQRdpKysjHXr1rFu\n3TquueYavvWtb1nvCwoKgNDTsMZRtCdJRyL47W9/28kmoicxthrZIJt9C0JfQBJBD7Nt2zYqKyv5\n0pe+RFVVFbt27Yo4pV21ahVXXXUVAPv37+dzn/scNTU1zJkzh7feeqtTex988AGzZ89mxowZTJ8+\nnbq6OpYvX87mzZutp5mPHj3K4sWLmTlzJtOnT7ceUtu2bRvV1dVceeWVVFVV8elPfxqfz8cjjzzC\nunXruPDCC60zmIULF1q+Pg899BDTpk2jurqam266CQgdTEtKSli+fDknnHAC8+fP58CBAwA8/fTT\nzJ07lxNPPJEzzjjDmv6DH/yAL3/5y5x00klcfvnlLFiwIOIBG6e1NYSsp08++WROPPFEZs2axerV\nq4HQU81/+ctfrPkuueQSnnrqKQKBADfccANz5sxh+vTp/N///R8AL730EosWLeKss85i2rRpAJx9\n9tnMmjWLqqoqaz6A3/zmN0yaNIm5c+dy1VVXcf311ye9fQShP9A/bhZny4c6Bps2beLBBx+kpqYm\n7q/Rf/u3f+N73/se8+bNY8eOHZx11lmdnkT81a9+xXe+8x0uvPBC2tra0Fpz2223sW3bNuvA7ff7\neeqppxg0aBAHDhzgpJNO4qyzzgJCzqQrV65k2rRpfO5zn+Opp57ioosu4he/+AX/+7//G2GjDbB7\n925+8IMfsGbNGgYPHszpp5/Os88+y7Jly2hsbOTUU0/ltttu44YbbuC+++5j+fLlnHLKKZxzzjko\npbj77rv56U9/yu23326Nxeuvv47H42HFihXcf//93HHHHdTW1qK1pqqqKqL/UaNGWdbZmzZt4rLL\nLmP16tWWDfbSpUvx+Xy89tprrFixgnvuuYfhw4fz9ttv09bWxrx58zjjjDMAWLNmDbW1tYwfPx6A\nBx54gNLSUlpaWqipqeHzn/88zc3N3Hbbbaxdu5YBAwawaNEi5syZk/T2EYT+QP9IBL2M4447jpqa\nhE9189JLL0Vclz9y5Aitra0UFRVZ0xYsWMCtt97Kzp07+dznPkdFRUWndrTWLF++nL///e+4XC52\n7drFoUOHAKioqLB+Ec+aNYsdO3bEjWn16tUsXrzYUjN88Ytf5PXXX2fZsmUUFRXx6U9/2mrrb3/7\nGwAfffQRF1xwAfv27aOtrY1JkyZZ7RmLaQiZ1p144oncdttt3HfffVx66aW0trbicrnIy8vD7XbT\n1tbGtddey/vvv4/b7bYsAs4880y+/e1v4/f7ee6551i8eDGFhYW88MILbNy40fJDamxsZOvWrQDM\nnz/fSgIQcmN95plngFDC+/DDD9mxYweLFy9myJAhQOjMw9hoJ7N9+gJGKWPsC+w1f3tKmeJUCgER\n/SVDLHVPT1wvNwZ0idqzx9Ad6uvrLdsI+zRzA9hY+7S0tKCUoqioKEJlFAgE8Pv91NbWUlBQEPV7\n35P0j0TQy3yo7eZuLpcLu5+TXSKmtebtt9+27ilE49JLL2X+/Pk899xzLFu2jPvuu4/Ro0dHzPPg\ngw/S2NjI2rVrcbvdjB071urHboGdl5fXrevl9jjtbX3961/npptu4jOf+QwvvfRSRO1m+1gMHDiQ\nRYsW8cwzz/D444/zj3/8A8C6j+J2u/npT3/K2LFjWbFiBR0dHZSVlQGhg8nChQt58cUXeeSRR7j8\n8suB0Bj+6le/4rTTTouI1RT2sb9//fXXeeuttygqKmLhwoVxbbBN24m2T1/AngSATqUceyIROPuI\nhUkUzv/BYNBSypSWlnZbLeTs0xyA47VXUlJiqZe6OyamP3tfdtmqHaVURIz26em4xxgNuUeQZlwu\nF0OGDGHr1q0Eg0GefPJJ67PTTz+dX/7yl9b7aN77dXV1VFRU8M1vfpOzzjqL999/n+LiYpqamqx5\njMWz2+3mxRdf5OOPP04Yl7MNw9y5c3nllVc4fPgwgUCAVatWceqpp8Ztq7GxkTFjxqC15oEHHog7\n71VXXcV1113HggULLJdRZ1sjR45EKcXvfve7iCR64YUXsmLFCv7xj39YNROWLl3Kr371Kyspbd68\nmdbW1qjtlpaWUlRUxIYNGyzL7jlz5vDKK6/Q0NCA3+/niSeesJZJZvsI0fF6vVHPBEpLSyMUQOZ9\nNAVRd9RC0fo0f7Hai6Yw6g5er7dTX0Y+qpSyxsjj8VjzmWlut9uSnKb7bAAkEWSE22+/naVLl7Jg\nwQLGjh1rTf/lL3/JG2+8wfTp06msrOTee+/ttOzDDz9MVVUVM2bMYMuWLVxyySWMGDGCWbNmMW3a\nNJYvX86ll17Km2++ybRp01i1ahXHH398wpiuuOIKrrrqqk5y17Fjx/LjH/+YRYsWMWPGDObNm8eZ\nZ54Zt62bb76Z8847j9mzZzNixIi4886dOxev18sVV1wR9fPrrruO3/72t8ydO5cdO3ZEnNEsW7aM\nv/71ryxbtoz8/HwAvvrVr3L88cczY8YMqqur+drXvhb1rOfMM8+kpaWFyspKfvCDHzB37lwgVFL0\nu9/9LrNnz2bhwoWUl5dbCSqZ7SMI/YG02VArpcYBDwIjAA3co7X+f0qpm4GvAAfDs96ktf5TvLb6\now11rrJr1y6WLFnCxo0bI06T7ZXc2trarGvY9kSQLowNtt/v59xzz+VrX/uaVTIzGXr7vmhsjqOR\nrA10sn04f1F31Wq6q7bT3aWn+o3WjpkGdLpXYV9nJ92JJVkb6nSeEQSAb2utK4F5wNeVUpXhz+7S\nWs8I/8VNAkL/4be//S0LFizgJz/5Sacyndnkhz/8ISeeeCLTp09n8uTJluJKEHKFtN0s1lrvBfaG\nXzcppTYCY9LVn9D7ueKKK2JeEsomd911V7ZDEISskhHVkFJqAnAisBo4CbhOKfVlYA2hs4YjXWlX\na92rfln2BQKBgFW72CgSXC5Xxi7F2C/72AkGg7S1tUXEZaYZaWkqfQAZuazU2yv8GRljLIys0i4n\nbWxsBGDixImd2jIS0Vg3XJ0+QqnS0tLCli1bCAQCPW6zkEz8JoZEEtJobZlpgUCA9vZ2XC6XJQk1\nqqqCggICgYClkoLQfr5582b8fj8ulwullPUddblc7N27l1GjRvXUMEQl7TeLlVIDgceB67XWR4Ff\nA8cBMwidMfw0xnJXK6XWKKXWHDx4sNPnHo+Hw4cP9/ovYm+jo6ODYDAYIUuzH3jTTbS+lFIRicH+\nOhgMWl+KVPrIxLporTl8+HDGfWFSwS6bNDgN0Hw+X4TxW6y6ucnUGTZtlpSURNQRTgYzX3t7e4/I\nRp0kE7/TpC6Vtow81IgvgsGgZbxnknEgEEBrjcvlwu12W8cvv98PRNrSmHkyQVp7UUrlE0oCv9da\nPwGgtd5v+/xeIGrRXq31PcA9ELpZ7Px87Nix7N69m2hJQoiNXVET7WCZbs28+ZIk2snNr6Zk5o3V\nRyb0/x6PJ0IJ1htJdLM23s3k7vaTivSztLTU8r/K9E1iewzdcf00iTXeD5GioiJr/cxDbuaMzCxv\nnycTpC0RqNA1mxXARq31nbbpo8L3DwDOA7r0zH5+fn6nU1chMXZlQrQvf7rVL8aiIZWDU6rKFtNH\nb1byCEJvIp1nBCcBlwIfKKXMkzg3ARcrpWYQkpTuAL6axhgEQRCEBKRTNfR3INqdXJGLCoIg9CLk\nyWJBEIQcp3+YzglxMbI2I1mL56di7iH0dO1W03+y89ufOm5paWH9+vWWAiVaXHZnR/u0TNV8TYQ9\nvkTr0JOOoNGe9o2HfdztTqCHDh2ybsK3tLRQW1uL2+3G7XYntV8544q2nvX19dY+4tx2qYzN9u3b\nOXbsmBWPUQKZe05GMmvaita2vZayXVp77NgxmpqarDhbWlrYsGFDTPWi8z6c/Ttg9nO7kq61tRWt\ntRUjhG4u93n5qJB9jHzNfMmNtM/glD+mo3arvX8TQ7z57SqKZOKK5uyYyZqviTDxJVqHRPLGVPuE\n+GNtPrfLSZ3LNzQ0WEnAfB4MBmlvb7fWKZl+7O1GW0+nFDOZZaJhpK/28bbLaI0yx75+9rbNehhJ\nrV1aa08ChlhJIJas2C5RdS5r3tufM8gEckaQY9gdEWP9sorledJT/SerADLuk0YFlEivb+aH9K5D\nV8nG8wbRHDCdlJaWdpon2vi5XC4qKiqora2NSNTmrKMnzmJ60v3TOd72fS/e/mEkpLEkteYsIREV\nFRURhYzMeNnHqaioKEId19LSYrVv35/TjZwRCIIg5DiSCARBEHIcSQSCIAg5jtwj6Eds374dCJmF\n2VUqXbEPMMZbQJeULNFq2Ca7nFPdYTAKi/Xr1zN69OgIsy+nOsYYf9XW1lJYWGiVA4yl1tmzZw9A\nRLvdJZZaK5qpmX29nZ+nqibqTr1foxAy3jf2a9zBYJBt27ZFqFxSKX1qjytan+3t7RQUFOB2uzvV\nL7bvE7HUYNGUY5B87eREZnOpfo8+/PDDiPf28eqKoiudyBlBP8JuFmZXqRiSNQCzqxq6qmSJtlwy\n/UdTd7hcrk436KIpTKK1bSR58dYhnlqlO8RTa8VSzJgDtzOmVLZBd+r9GoVQLGdfsy72z5Ldr2Il\nJ7sqCTobv9nX2xi5xWvf2U6s97HaMPN25SBdVFRkvbaXTDVGiiYRRNtnTZ/FxcUpmfX1BHJG0I8x\nX7hUqkRB9423opFK/7HUHfF+kTlVK0bjnsgALFPY44s1tma9e0Lx1BOKE6MIMrFB5L7UlWpeJi7n\nOppE73a7o+5/Xq83qW3pXG/TTiJVk/Ogb5RU9jMiJ3Z5s0lAxx13XFQTP/Ojxtmn3WgvmnorU8gZ\ngSAIQo4jiUAQBCHHkUQgCIKQ40giEARByHHkZnE/wUg94ZOqR921NHDe8DLyvPz8fPx+f1w5pl3u\nB8nZKzgldU5TLmds9ht5phqZU4poX7a1tZUNGzYAIXWHUXDYJZD2du0md045rF2hMnLkyIhxMKZn\n8TBmZfn5+ZZc0l6i00gZ7WZvdjmtPQ5njPa27BJWY5wWbZvt2rWLpqamCO8b+/YPBALWOBkJp9k2\nxpwtFYmxXd5r30Z22XJLS0tELV+70dv69eutynWBQMCq86uUYsOGDbhcLkaMGJGw/w8//JDW1lZr\nvILBIBs3biQvLy9hGdxgMNjpOxLLgM4+n32/tb+ur6+nsbExwjAPMmM6J4mgn+BUWTilg6lK0UpK\nSjrt5PaDjPlSRvvi26WQ5guWqnTPvLcfJKIpL6K1YZci2lFKRbRn5jMeL07MtGjeM/bXDQ0NEeNg\nTwJmDJzboqWlBa017e3t1gHWxGY+d5q9mbG0G6Y5Y7QXRDexOdct2jYzZmoulyvmAdDU2tVaR2wb\n+zZJVfUSbfua5NbS0mI9zxDtx4AxvnPGCKHa3PbEHWv/MxJPrTX5+fm0t7fT0dGRcp1sZ/9dwb6t\neuKHXCpIIuhHxJJodkWSZpe1dWWHTLW8pH25aPEaw7NYclJnTWPza9F+oLDLIcvLy6mtrbVeG3qq\nfq9Zl1jbY9++fZ2M2+yfOxN7RUVFRIyxcLaVKtFq7ppfp263u8fkrc5t49y+iYzf7MvZY3W+jyXJ\ntPdvtoFZt2T6hOjJKZohnV16m6hNQyYN50DuEQiCIOQ8kggEQRByHEkEgiAIOY4kAkEQhBxHbhb3\nUeyOmcaHpSdVBvb6sXbpIBAh49uyZUvEDTJTcSnV+rUQqZTYtWtXxM21YDDYSWpoxyknjSbts7+v\nq6tDa43Wmi1btlBUVITf748wCrOvo1n/WOqiXbt2MWDAAA4dOpTUOjvr1BpiyQ+NK+aWLVs6KWVa\nWlrYtGkTgUAg4qajkRE75zXyTBPr0KFDrc+jyXVNHV0j0TXzOTF1fs3yQ4cO7eQQW1BQEDUuI9vU\nWhMMBqmvr0/K2dQZazQpqtPl1S7JjdZGIuIpg2JJnaMZ+DmXs+8HUrNYSIpEctF0tG+XMBpM3dr2\n9nba29tpamoCUqtfa3/WwCxn2rF/gZL9wtqXifUFtB+ITNxG0ukkEAh0qufrpKmpKaaLphPn2Nr7\njHWQMcvEksaag6ZzeRO3qUls2jKxtre3R8Tj8XgoKCigoKCA/Pz8qG2WlJREjIXTfC1au/bXJuE7\nt01HR4fVl33+/Pz8iP0uLy8v6hhA5+0dzeU11hiadUtEPIltd7DH7lTBpRs5I+gHpEtmZv9Vn4ob\naKr1a+NJ5ew1XZ24XC4qKys7OUQWFRVZBxt7zWN7TVgjk0z0hbZLC+GTB4DMgSnesw3JuF063T2T\ndUs1Z2F2KaMz1mjEk366XK5Oy9rllObgZJdk2h1IE0kvY8lQzTrbt4uZzz6v2XZTp06NeG+XzFZV\nVVmy4FiJ296Pc6zt0l7nMpWVlVbMpj/7drD3Z86KzXzGntr5nIc9wdmXz6R0FOSMQBAEIedJWyJQ\nSo1TSr2ilKpVSm1QSn0zPL1UKfWiUmpr+P+QdMUgCIIgJCadZwQB4Nta60pgHvB1pVQlsBz4q9b6\neOCv4feCIAhClkhbItBa79Varw2/bgI2AmOAc4EHwrM9AHw2XTEIgiAIiVHpuPvdqROlJgCvA9XA\nR1rrkvB0BRwx72NRU1Oj16xZk+4w+wx26ShAdXV1Wtq339AsKCigqKiIxsbGuF4rEHICnTRpktVW\ntMLrTvmruTnmlPdF826xo5SKecPXtGu/wWvairdcLIzTpSHeTUf4xCjO7vxp1i0Z/xkn0cbCuS1c\nLhcjR44EiDrudulmrBuidjZu3NjJgG306NERN4vN/gGdVU0FBQWdTNyM6ibZ8XeWqTTut9EoKCjA\n7/dHtJ2Xl8fAgQM5evSoNT3a9s/LyyMYDHZZEVRQUGC53ppt3BXfJ+N6e+jQIQKBAAUFBRFeU6mg\nlHpXa12TaL603yxWSg0EHgeu11oftX+mQyMeddSVUlcrpdYopdYcPHgw3WH2KeySuAEDBqStfXux\ndbs0VGsd1eQtmlQzVuF1+3u7ZM8p70v0bES8L61pt6CgoJOqw46J2+VyRSilnM9C2JNAQUGBNQbF\nxcWd2oKQQsQUr3e6idrX2SnJjYX9wG0OvG63O6J/4wwaa9yN9NM5Fi6XK6IdZ5/29Yq27eyJ245x\n87STShKAT+SmRjYaKwmY/pwxdHR0dLLYjta/Xb7aXeIVnzdjqZSKeDbDUFJSYu0rmaq3ndZEoJTK\nJ5QEfq+1fiI8eb9SalT481HAgWjLaq3v0VrXaK1rhg0bls4w+yRer5fq6momTpyYtvaNTND+xXK5\nXBQVFVm/+M28lZWVVFVV4fVSyEhvAAAgAElEQVR6U9ZAx5JZml/0Xq/X+ouG8wBuppl2J02aRGVl\nJeXl5da6FBUV4XK5cLlcVtxGelldXU11dXXE/E4mTZpkree4ceOs6VVVVVRXVyd8oM7EZsbOvs72\nZaP17/V6mTRpkjXW9v6NTDFev+Xl5RHrZmKwt2NQSlljFKs9047b7bYSjNkm0eK3x2i2QbzxMtul\nsLCw075o+rD3VVhYGLc9M1+02JJNytFwu93WNikvL6e0tDTqvmv2N/M9cvZp9g2zXl09G0iFdKqG\nFLAC2Ki1vtP20TPAZeHXlwFPpysGQRAEITHpfKDsJOBS4AOl1LrwtJuA24A/KKWuBHYCF6QxBkEQ\nBCEBaUsEWuu/A7EMNk5LV7+CIAhCaojFRB/AWS83FVO3ZDF1do2iI9q1ZntpQnuNZGPMNnToUKsW\nrKkpa5ZpbW2ltrYWrXWnkpH29bSrLIzxVqL1jWYwp7W2FEim5m1+fn5EPObGYG1tLUopCgsLrWU6\nOjqSUpAYozWDMcaLdq3ZaXoXzSgwmhmbk9bWVmt5Y8ZnaGtri7g5awzxWltbCQQCFBcXM27cuE6m\ngnV1dRG1j41iJdq62I3r7CUr7WMaDAbj1pu2r5/L5YprMGf2L6ciyWx3l8sVs4yofV7n61jlL7uK\nMZfTWrN+/fqYyjRT79ntdkdsB4PdMsXn87F3716pWSxErzvbkwZz8EmdXbvUMRbBYLCTGsVpMmbm\ng9DNPrsCwv7lsKue7MubL5H9wFhSUhJR3zeRO6VdpeOUMCqlLBmi/YuYyJQMPlHsmLq69hjtBxmv\n12uNpzkw2olmFGj8dbTWtLW14Xa7O8khzbjYD7rO/s06aq2tesTwiZmfc1s5ax/bVUBGTWRkkfbl\nfT6fNd150EtWgVNcXNzJ9dVJvG1SXFxMY2NjUn11lWTlxskaCJr917kdnGRKNSSJoA+Sqqlbqng8\nnqTat//itytH7F9ao/yx/4I18zs162a681ey/SDl1MRDpGmZU+7o1N47jfQSGaVF0+4bxZS9rm60\nZwqcxmHmbCFaLWMzfnY1ln09nQZrZoyimdaZA5Dd6Cze8xjObWbicRrYTZo0qZOFtL3Wr/MMwG4Y\n6DSHs5v2jRs3jrq6OkuF5TR0c752Mm7cOCsRdFW7H41o9urR1sPuvBorRtNWtDPbeIaDXq837WcD\nIKZzgiAIOY8kAkEQhBxHEoEgCEKOI4lAEAQhx0nKdE4p9SngeK31S0qpIsAddhTNCLliOhfNoM1u\nzma/wZZMBSPTnlHXmLq8dpmgUS04b7CNHj2aY8eORShOukK8G35GfWNq5pr1zMvLo6Ojw/LzMTfp\nKisrO0kbu4PpJ1WUUpaax040A7jCwkJLLmu/yTp69GhrPezySWPoZraN3++P6MfE7PV6LalutL5T\nIdY2Mje1TSzxbsImUtUopaLKhgsKChg6dGgnk8Ou0tVtmgr2dTXjZVdUdbdNJx6PJ/umc0qprwCP\nAb8JTxoLPNWlqIS4RDMKM6+NUVgqtYlNPWB7PWF7+/Z6wQZTp7ahoaHbSQAiD0zFxcUUFBRYZlv2\n2rb2dbYvY9bVTItmxpVMjeJomANGPG8kuymYaVtrHdX4LBgMRhi3mYLkTompcz3sMlj7tmlpaenU\nj1Ge2KW05gDb1W1lr0Vt2igoKLDGPlESgMRS0XiqH6fJYTQjtmRJp9wy2r7lNOpL1qvI2VYmXKDj\nkYx89OvAHGA1gNZ6q1JqeFqjEiKIJjdMlng2yeZze/1Y+2uI/4ve7PSVlZURD1LFmt9pamaWsbdn\nryXsdrspLS2NsNyORlVVlfUQjqlX7PxVG03eaPqcMmVKRF1j+MTxMpYE1I5dsjhu3Dj8fn+PSRhN\njEayqJSyYtq3b19UOWOyB0P7fmUfn0T7W1d+uRcWFkac2ZnxNknYLokuLS3tVBs4WRKdmdjlnrHk\nw7HknM6HLM0YmWRt/0GRKG7nfmpii7ZvZaJ+cTLpq01rbZ3zKKXcxLCOFgRBEPoeySSC15RSNwFF\nSqklwKPAH9MbliAIgpApkkkEy4GDwAfAV4E/aa3/Pa1RCYIgCBkjmXsEXwJWaa3vNROUUmdprZ9N\nX1iCIAhCpkgoH1VKNQA7gIu11hvD09ZqrWemP7wQuSAfddYhdrlc5OXl4ff7LZWIXV3hlH/Gqgkc\nq86tuSlrVEhGvhfPfyfWjbhY/i7O+Y3kMVlnTyexboRWV1dHrUtsX1enK2e8PgBLEjh06FBLgpuM\nZDVazdxUiLWOyXropFqLOS8vj8LCwk4+UCNHjqS0tDRqzeJ0oZSKEAt0V5KZCZz1lNNBWVlZl/2G\nerJm8XbgX4HHlFJfMO13KSohJtGcO40awSQBn8+Hz+frJP+0T3O2F03lYzTtJgnYk4n9vx1TsjAa\nTmmkHaPAcLlclpS1q7Vhoz2DYDAuph6PJ0KaZ5w6k/2iaq07uZH6fL6kn1uI5sKZLM5awvbXJSUl\nVs3eeKRaarGjo6PT+NjdZTOVBCA09nYr654kFVlxKthN/rqCfZs6t12iEp49STJ7jdZarwVOBa5W\nSt0BJN4jhZSJVkvVLquLVnQ82jRne4bKyspOD6XZa+fa5XvO2rBVVVVWLdjq6uqIz6MVQjfF7E0N\nYBNnKnVi4013unROnDjRqjNsl/kVFRVFJAcTW6x2zbIul8uSA8Yq9g6f1BiOVwc3GVwuFxUVFRFS\nQftDRKWlpUydOjVhO4WFhVGnx6uDHGv/SZbq6upuLR8L5/Md8eogG2KtZ1VVVcJtE+tAHK9Pj8fT\nqQZzvPad9YunTp1q7ZN2N167+2tvcR/dC6C1PgQsJSQdTc+WFwRBEDJOwkSgtT7T9jqotf6u1lo8\nigRBEPoJMVVDSqmfaa2vV0r9kSgPkGmtz0lrZIIgCEJGiKkaUkrN0lq/q5Q6NdrnWuvX0hqZjf6m\nGnKay6Wi8IHQ4+uDBw+msbHRMvMydYbj3Yx12j8UFBRYN3GddgobNmyIMNYaOXKkFaOzNGA8pYqp\ngdzTNwDtihKjRgIslU9XLR6imZZ1x9AtFQYPHgxgja19XN1ud7f8hHKddJnRpWLpEQ37No71PcqE\n6VzMMwKt9bvh/9YBXyk1BBintX6/S1EJQKSJlyl3CNFLBtpvWNqL15vas+Ygaw7o8XZK+2dGTWMS\njVMpZN8hi4uLI1RIpu9Y2HfodB24nOPkrKXbVZx+SYCVZIPBIG1tbWkzCHOOq72fRPWZe4qeLPXY\nm0iX+ine/p2MlDeZGseZIBn30VeVUoOUUqXAWuBepdSd6Q8tt/B6vbjd7ghlgdfrZdKkSZSXl1sK\nn9LS0qiSsljqITOvXf3g9XotpYNTPeTE5XJZZnFONZN57zTjqqqqSmqdoyks4ik0YqlzysvLOy2T\nSF0SSxFitOx2dUdRURHl5eVUVFRQVFSUUHlkjyEZEqlg7BJc53SzHtGUO9EUNImUM4kMzhIt35Ny\nx66MpX1+o27rqpIrXtvJTE9WTWRf3sRr315dPRtIhWRGaLDW+ijwOeBBrfVc4PT0hiUIgiBkimQS\ngVspNQq4AEjaVkIpdZ9S6oBSar1t2s1KqY+VUuvCf5/pQsyCIAhCD5JMIvhP4C/ANq31O0qpcmBr\nEsvdDyyLMv0urfWM8N+fkg9VEARBSAcJTee01o8Ssp427+uAzyex3OtKqQndCU4QBEFIP0nVLO5y\n46FE8KzWujr8/mbgcuAosAb4ttb6SKJ2+ot81MhG7eoeIw21K1Xy8vIsNVB+fj6BQMCqfWv8WJLF\ntOusZGX/3NTfjVczNZ6fkJPumIV1V45nSEaxYeaJVRnKXhM4W3RH9tiVZVM1rRPST15eXlL2ItHo\nSdO5nuTXwHHADELWFT+NNaNS6mql1Bql1JqDBw9mKr60YpKAXeFjDrD2L5/RiyulLLMzU/s2WhKI\np0gwB7hgMBjVTE5rTSAQiPvlTyUJQPekjj11EEqmHTOPs+4sRDfe60mSMUErKCjoVlLsyrKSBHKT\njCYCrfV+rXWH1joI3EuoFnKsee/RWtdorWuGDRuWuSDTjJFrVlRURMgCjSwRPjlIxDOUM8t5vV4q\nKys7SQyjORdGk4gaU7Z4fTjfOyVuzs/t7TnN66K1aaiuro4wfeuqFDEZqZ49lnHjxnVaprS0NG5R\ne7NstHGIts7Oz+0S21jxTpo0ydoXuiJFtScb5/LdNcnrLvb+E8l8nTiN29JFrG3Xlb67sy939Wwg\npX4SzaCUGqGUWqGU+nP4faVS6squdBZWHxnOA9bHmlcQBEHIDMn8HLifkGpodPj9FuD6RAsppVYC\n/wAmK6V2h5PH/yilPlBKvQ/8C/CtLkUtCIIg9BjJlKocqrX+g1LqRgCtdUAplfAOlNb64iiTV6Qa\noCAIgpBekjkjOKaUKiPsQKqUmgc0xl9EEARB6CskU7N4JvALQsVo1gPDgPMzaTzX1+SjRiZqZJv5\n+fn4/X5LfWPq4faUNDHZtnqLoVhvkygOHjyYo0eP9qqYhNj0tv0n3SilkvbvirJs99xHDVrrtWEr\n6smEahVv1lr7uxRVjuC0QbaXhzTTe1KfHq8t55fGKEny8vJiyjxz7YvW1NSEy+XKaH1eoevk2v6Z\nCZJRDX0Z+CIwC5gJXByeJqSA0+8/2bqryRBP0lZUVER1dXVEPd+ioiKmTJkSsz37rw+7O2e8urfJ\nxGfaSqWQeKaKd8eq9ZsMXXEP7ck+Es3rdLPMtmQ0Hs562PZl7T+oUmnTtJvqMsnQlbaSca6109Wz\ngVRI5mbxbNtrD3AaITvqB9MSkSAIgpBRkrk09A37e6VUCbAqbREJgiAIGaUr50jHgIk9HYggCIKQ\nHRKeETiK17uASuAP6QxKEARByBzJyEftxesDwE6t9e60RuWgL8lHTSF6J0opRo0aFfUzJz1daNvc\noA0Gg0mpLXpS2pqJdgWhvxOtFGky9Jj7aLh4/Q4gX2v9BnBYKdXZrlEAPime7vF4IlQAWuukCqv3\nhA2zU5UTDAbp6OhIWnKXqULpgpAKqajNhNRIRj76FeAx4DfhSWOBp9IZVF/H6/VSUVER1dUzkXTM\n4/HE3OFjyTedksBoRbOTlbl5vV6reHt1dXVc+Waq0s5Ebp5CciTrgGmX/qYzFmef6erHvl876Yl1\nzZasNhFdPRtIhWS+mV8nZBe9GkBrvVUpNTytUfUU2z6C5sw+STvKePcf3cQon4+g7RKPqzlUW4Bo\nv8yVAq3jzuNqDnzSXnh+6zVY76POZ58/Dq5m29lAlHWIGU8SpDq/EAOlcLnaE45l3P2tB2Oxt5+2\nbZxgnXtkXZP8jmScbR9Bxfi0dpFMCmzTWlsXdpVSbj65eSwIgiD0cZI5I3hNKXUTUKSUWgJcC/wx\nvWH1EGnOotHYW1cHQHl5OXvr6iKsJrxeb0SZSjvm3kC8eexeQfZ7CeaU1ryPNl+y9x7sp/bR1iFW\nPMnQW7yO+jqxyo46ibcv9WQs9vbTtY0TrXNPrGtPlUntaaozcBxLRjXkAq4EziDkNfQX4P90Bs0+\neotqyGkmV1JSYlX9qq+v59ChQ7S3t5OXlwcg3jWCIHQbj8dDRUVFl5btSdM5U1by3i5F0o9wmsnB\nJ+UfGxoaLGlksjJNQRCE3kAyqqHtSqk6518mgutrOGsO26d1t01DNHVEb1U7dAW7UskolwzdVaRk\nYpy6aszXHfrC9u8LMfZWuno2kArJ3COwn1Z4gC8AnaugC4IgCH2SZB4oO2z7+1hr/TPgzAzEJgiC\nIGSAZLyGZtreugidIciTQYIgCP2EZA7oP7W9DhCym7ggLdEIgiAIGSehfLQ3kG35qF0aasflcqG1\nFoWQIAhpJd2mc8lcGroh3uda6ztTCawvYpeGQigBFBQU4HK5IuSkUktVEIS+SDKarhrga8CY8N81\nhGoXF4f/cgJ73V3zgIe9BrHX66WoqKhHasKaNnrCSCtWG11pN1P1g6FvyQ37Uqyp0BfWK1MxZnLf\nt+NyuXqN6dxYYKbWuglAKXUz8JzW+pJ0BiYIgiBkhmTS6QjAfnG8PTwtLkqp+5RSB5RS623TSpVS\nLyqltob/D0k9ZEEQBKEnSSYRPAi8rZS6OXw2sBq4P4nl7geWOaYtB/6qtT4e+Gv4vSAIgpBFknmg\n7L+AK4Aj4b8rtNb/ncRyrwP1jsnnAg+EXz8AfDalaAVBEIQeJ6kHw7TWa4G1PdDfCK313vDrfSRx\niSlbGKfRkpISAoFAhD1tS0sLmzZtinAX7UnrXdNXT7QZq42u2O1m0kK6N9oBx6IvxZoKfWG9MhVj\ntuzTg8Eg69evT/sN46w9Iay11kqpmFpLpdTVwNUA48dnvq6A3WnU1PAtKSkBQjtFpur6iiRVEIR0\nk2l92H6l1CiA8P8DsWbUWt+jta7RWtcMGzYsYwHGwuVyUVpaSmlpabelZHZJp5GH2V03nfWHM1F7\ntifpS7EKQjZJ5ruSCflopr+xzwCXhV9fBjyd4f4FQRAEB2lLBEqplcA/gMlKqd1KqSuB24AlSqmt\nwOnh94IgCEIWSds9Aq31xTE+Oi1dfQqCIAipk/N20kYdBETUIDaYgthKKerq6ggEAp3M51LFrkAI\nBoNs2LABpZSlgHAqlPoafUFtIgi9gWS+Kxs3bmTq1KlpjSPn7+o1NDTg8/nw+XxWQrDj8XisGzo+\nn6/H1EL2cpZaazl4CoKQNXI+EUDoYO/xeDpN93q9lJeX4/F4UErFnC8exjzOiTGos89n7zdbJleC\nIPQu0n02AJIIBEEQch5JBIIgCDmOJAJBEIQcRxKBIAhCjpPz8lEIeQn5/X601mzZsgWA9vZ2CgoK\n2LBhg+X10xUpZzAYjLqcc1pfl4wKgpAe+rXpXG8iEAhYB3un06gYvgmC0N+RS0MOzHMDLpcLt1vy\npCAI2aU/ms4JgiAIvQxJBIIgCDmOJAJBEIQcRxKBIAhCjpPTiaC+vp6WlpZO0s1gMBhT9ikIgpBJ\n1q9fn/Y+cjoRRHMbFQRByDVyOhEAllS0r9UFFgQhNxD5qCAIgpB2JBEIgiDkOJIIBEEQcpz+7aFw\n/fWwbl2nyf6wydwIKQ8pCEIv59CUKQz93e/S2kf/TgQxCAQCaEkCgtBnaW52ceSIm6amPAYM6GDc\n+HY6Aor8fDGJ7Ar9OxH87GdRJ++tq8Pn80nBeEHIEH6/4uc/H05FRRtTpviYPNnHvn35vP76QPx+\nRUODm4suOkxZWUfMNpqbXezbl8+jj5by8MNl5OcHGTAgSMNBN+4jQTo6FKee2sQppzQxcqSf5uY8\n5s9vpqQkdptdXx+or3czYkSgR9vct6+AkSPbcbth9eoBjBzp56yzJvVYH7Ho34lAEISs8dJLg/j4\n43wAdu8uYNWqMuuzyspWamuLIub/858H8+UvH6K+3s2gQR1MmNBGZaWPkpIO3nxzIN/4xnja213k\n5WkuuOAw3/72fjyeIPfcM4y6ukIA3n/fy6uvDrLaHDOmnUsvPcSwYQE+9al2Kip8tLUpVq4so6qq\nlRkzWigo0DzwwFDefHMgV111kNmzj2FXku/Zk8+773qprS3iz38ezIQJ7WzZ4qGpKY8LLjjM9763\nj8JCjdagNaSiQg8GQSnYuNHDt789nt27Cxg1qp3S0gAbNni5+OLDnHVWFwY/RVRf8NuvqanRa9as\n6bH26uSMQBDSRiAA9947jF/9akTE9KVLGzn99Ea2by/k978v41/+pYkvfekwbrdm584Cfv7zEdTV\neSKWKSvzM2NGC6+8MohRo/xceukhliw5yvDh0X+Jaw3bthXS2JhHa6uLn/xkNLt3F1if5+VpOjqU\n9X7AgA5Gj/azdesn/VZU+Jg0ycecOceor3fzm98Mo60tdHQvL/dRV+dh+HA/s2Yd489/LmHiRB+L\nFjXxwguDOXjQzZIlR6mqaqWhIY8PPyxkwoR2Zs48Rk1NC17vJ8ecxx4bwl13jcDj0TQ05FFWFuBL\nXzrMyy8PYs+efL761YOcfXYDs2ZVdXlbKKXe1VrXJJwvG4lAKbUDaAI6gECiQCURCEJ2+eijAtat\n83LggJtZs1qorGxl0yYPH31UyNSprezfn8+TTw5h9Oh2nn9+MHv3FnDWWQ3ccMM+/H5FU5OLSZPa\nUOFjsNZYrw1aw+bNHoYO9dPQ4OYf/xjIqlWl+Hwuli1r5OqrDzB4cOrf2X378jl8OI/a2iJ27y4g\nEFBMnuyjsDDIK68Moq6ukHPPPcJ55x3h6aeHsGLFMDo64NCh0NnMaac18rWvHWDYsAClpR1s21bI\n4MEdDBsW4I03BvLf/z2KnTsLOfHEYxx3XBvPPVdCa2socQweHODYsTwCAYXLpRk6NMDcuc0MHx5g\nxYphzJnTTFFRkPx8zfe+t49Ro/yd4u/OA2V9IRHUaK0PJTO/JAKhP6A1bNhQxPPPD+bDDwspL2+j\nowNqalpYuLAJjye930W/H7Zu9fDqq6GD35lnNnDqqU24XHDggJstWzzMmnWMoqJQHMeOhS7DvPpq\nMf/xH2Noacmz2lJKo7Xq1IdSmlmzWrj88kOcckpTp4N9X0FrWLNmAAMHdjB1qi/hvIcPuxk6NHSW\n0tamaGgInZEMHx4gGIR33hnAypVlDBjQwdtvD+DoUTeLFh3lzjt3JbzBLYkgTE8lgvr6evbs2dPt\ndoTM4/dDfn62o+g6GzZ4uOee4bz8cuj69cSJPnbvLiAvD3w+FxMmtLFwYRNNTXmUlgasG6jmlzfA\n7NnHePbZEg4fdjNwYAdHjrhpaXExblw7X/7yIebPP0ZRUZDDh/PYt6+A7dsL8Xo7mDWrhRUrhvLU\nU0NoaAjdFvR6O2hpyWPYMHPppZhAwEVRUQczZrSyebOHo0dDv2QBxo1r4yc/2c348e2sXTuA2loP\nEya0M2mSj23bCmlocHP22Q0MHNhBXl70MRBC+P2waVMRU6a0Jr1PdzUZ9PZEsB04AmjgN1rre6LM\nczVwNcD48eNn7dy5s9v91tXViaNoL2DLlkI2bChi8eImBg+Orejo6ICHHy7joYfK2LOngIoKH+ee\ne4Tp01tpbnbR2uri4YfL+K//2s3YsZGn1MeOuXjvPS8jR7ZTXt7eqe3QLz4vhw7lc+qpTda12yNH\n8igsDOL1xv5e1NfnsWNHIR5PkMmTfdaBr7VV0d6urMsXBw+6ueWW0TQ0uHnvPS9FRR1ccslhli5t\nZPLkNiB0UHj11UHceutoWltdFBd3cORIHn7/J3ccy8r8tLS4aG3NY9SodjyeIMGgorKyFY9H88Yb\nAzlwIHREifVLHULX6I2i5sQTj/GXvwzmuedKeOutgcyb18xFF9XzyivFvPXWQKZObaW0NEBrq4sF\nC5pZurSxTyfivk5/TQRjtNYfK6WGAy8C39Bavx5r/p46I5BEkH3ef7+Ia6/9FI2NbkpLA5x8chOl\npQGeeaaEKVN8eL1B2toUbje89loxHR2KuXObmTGjhdWrB7Bu3YBObY4f38b11+/n5JObOHbMxTvv\nDOCmm8bi97soK/Pz9NNbrYOz3w9r1w7gF78YwXvveQEoLu5gwYJmPvywkG3bPAwY0MGMGS189FHo\nF/vkyT4OHgwpWUpKOvjTnwbT3h46UE+Y0Ma8ec1s3eqhtrYIvx8+//kjlJZ2sHJlKQ0NbkpKAnzh\nC/VceeUhBgxIfDkyGAxdXti8OXQmcMIJoRuPu3cXMGmSj8LCyO+sWad167y0tytKSwOMGeNn2DA/\n9fVu/vlPL6eddpSqquiXOAIBkPLcvZd+e2koIgClbgaatdZ3xJpHEkHfYN8+N08/PYSODsXgwQEO\nHsynurqVrVsL+exnG9i+vZBrr/0UQ4cG+M539vLMM0PYuNHDoUP5DBjQQWGhpr7eTX5+EI9Hc/rp\nR1mwoImlS49a15rr6grYvLmIoUMDbNrkYcQIP7/4xQh27CikoCBoHaBPPPEY5513hFtuGcPEiW18\n9rNHeP75wWzcWERHh2LYMD/XXnuAcePaeeKJIfzzn14+9al25s5tZufO0BnLhAlt+P2KrVtDyWHH\njkKUgrPPbuC0045y+LCbhx4qY+fOQo4/3sfUqa20trr4058GEwi4OOmkJr71rX3Wr39B6Ar9MhEo\npQYALq11U/j1i8B/aq2fj7WMJILs0tamKCzUbNjg4Y03iiku7mDfvnzKygLMnh26Ln3kiJsf/nAM\nO3cWRm3DXLIoL/fx0EN1FBeHfhlrDW+/PYCysgDjxrXT3q6sz5KloyOkQX/jjWJKSgKMHOnnC1+o\nx+vV/O1vA7n99pCqY/hwP+eee4SKijYWLz6a8s3ZlhZFfj6dbu45FTBNTS6OHXMxcmTPPWwk5C79\nNRGUA0+G37qBh7XW/xVvGUkEmcPvV+zalc/YsX6OHnXxy1+O4MknhzBypJ+PPy5IuPztt+9i6dJG\n9u/PZ8iQAPfcM5ySkgCHDoWewly6tJFhwzJ7gAwGYf/+fIYODYgFgdDnyEQiyPiVQa11HXBCpvsV\n4lNXV8DLLw/i8cdL2b27wLpR6Pcr5s9vZseOQr7ylQNcfvkh6+nOPXvy2bcvP3yTM8iYMe1UVIQu\ng4weHbp5+81v7s/magGhJz2j6bMFQQiRE7eI6uvr2bdvnzw3EIN33/Xyla9MwO93MWpUO9/5zl7e\ne89LcXEHV1xxiAkTnKqb0DgOGdIR8wakIAg9w/r169NepSwnEkFDQ0NOJoE33xzIm28O5PBhN21t\nigULmjnrrFCdZo9H09EBd9wxkoceGsqwYX5WrtxiM9E6nL3ABUHIKDmRCHIJraG21sPPfz6CN98s\nprAwSFlZAKXgxRcHc8stY3C7NaNGtbN3bz6BgItlyxr45jf396iToiAIPUMmahZLIuijNDa6+PWv\nh9PcHHqa6bjj2vjHPwayZ08+O3cWMmhQgO9+dy8XX1xPfn7IGfGFFwaxa1cBu3eHnjo9/fSjHH+8\nj898plGeBhWEHEYSQRBhMqIAAApsSURBVB8gGIQdOwrZsMHDhx96ePddLx99VEh9vduyCoCQa+K4\nce1ceulhli1riDDoUgqWLj2arVUQBKEXI4mgl6I1NDTk8dxzJTzxxJAIm9wJE9qYMqWVa689wAkn\ntHLoUMjgatw4UcYIgpA6kgh6AcFgSL9fX+9myJAAL7wwmDvvHMHhwyFzl8rKVm68cQ+TJ/uYMsXX\nyaZg6NAOQo7egiAIqZMTiaA3PkTW2OjihRcG89hjpezbl099fWhTmMIZ1dUtXHnlIaZNa2HGjNYs\nRysIQrYQ+Wg/IBiE9nbF2rVehg8PcOSIm8cfH8Lzzw+mo0MxenQ7U6a0Ul3dalk1nHRSE3PmHBMj\nMEEQMoIcarrJ3r0h6wKtQ7bDb7wxkMZGN1OmtPLHP5bwl78MJhiMtAUuLAzyxS8eZsmSo5xwQktK\nNU4FQcgtRD7ay9i5s4BVq0r529+K8XqDuN2aDz7wUlAQumZvnC8NLpfms589wp49BcyZc4wxY9oZ\nNKiD6dNbGDQo9x5wEwShdyKJIAHBILzySjHvvefloYfK0BoWLGhGa0VjYx5f+coB/H4VdtZsY/z4\nNsrKAhw9mseYMX6rfJ0gCEJvRRJBHA4dcnPjjWN5662BAJxxRiM33rhXDu6CIPQr+n0iWL9+fUrz\n79/v5tlnS2hszOPpp4fQ0uLi+9/fy8KFTVHM1wRBENKLqIYySCAA9947jPvuG4bPF7rWP316C7fc\n8rFlrSwIgtAfkURAqObrzTeP4ZlnhnDGGY1cf/1+ysoCVkFzQRCEbCGqoQzQ0QHf+c54Xn55EF//\n+n6uueZgtkMSBEHIKDmdCA4dcvPtb49j7doBfO97e7n0UvHgFwQh98jZRLBlSyE33jiWXbsK+Y//\n+Jjzzz+S7ZAEQRCyQs4lAq3hwQfL+NnPRjJwYAd33vkRCxc2ZzssQRCErNGvzQ2iSUefeGIId9wx\nin/5l6P88Y9bJQkIgtCrSVUC3xVy5oygvV1x662jefrpEubObeaOO3aJx48gCAI5lAhuuWU0zzwz\nhEsuOcQ11xyUJCAIQp9A5KM9QHu74oYbxvHaa4P4ylcO8G//diDbIQmCIPQqsvK7WCm1TCm1WSm1\nTSm1PJ19PfnkEF57bRCXXHKIr31NnhEQBEFwkvFEoJTKA34JfBqoBC5WSlWmoy+jEDrhhBa+9719\n5OfrdHQjCILQp8nGGcEcYJvWuk5r3Q6sAs5NR0fr1xfx0UeFfP7z9SiVeH5BEIRcJBuJYAywy/Z+\nd3haj/OnPw2moCDI6acfTUfzgiAIaSen5aNKqauBqwHGjx/fpTaWLm1kwoR2iovFPE4QBCEW2UgE\nHwPjbO/HhqdFoLW+B7gHoKampksX9y+55Ljwq9FdWVwQBCEnyEYieAc4Xik1kVACuAj4Ylp62vYR\nNLekpWlBEISMMNALFV27KpIsGU8EWuuAUuo64C9AHnCf1npDpuMQBEEQQmTlHoHW+k/An9LeUZqz\nqCAIQn9AjBYEQRByHEkEgiAIOY4kAkEQhBxHEoEgCEKOI4lAEAQhx5FEIAiCkONIIhAEQchxJBEI\ngiDkOErr3u/Rr5Q6COzs4uJDgUM9GE5PIXGlhsSVGhJXavTWuKB7sX1Kaz0s0Ux9IhF0B6XUGq11\nTbbjcCJxpYbElRoSV2r01rggM7HJpSFBEIQcRxKBIAhCjpMLieCebAcQA4krNSSu1JC4UqO3xgUZ\niK3f3yMQBEEQ4pMLZwSCIAhCHPp1IlBKLVNKbVZKbVNKLc9yLDuUUh8opdYppdaEp5UqpV5USm0N\n/x+SgTjuU0odUEqtt02LGocK8fPw+L2vlJqZ4bhuVkp9HB6zdUqpz9g+uzEc12al1NI0xjVOKfWK\nUqpWKbVBKfXN8PSsjlmcuLI6Zkopj1LqbaXUe+G4bglPn6iUWh3u/xGlVEF4emH4/bbw5xMyHNf9\nSqnttvGaEZ6esX0/3F+eUuqfSqlnw+8zO15a6375R6j62YdAOVAAvAdUZjGeHcBQx7T/AZaHXy8H\nbs9AHKcAM4H1ieIAPgP8GVDAPGB1huO6GfhOlHkrw9uzEJgY3s55aYprFDAz/LoY2BLuP6tjFieu\nrI5ZeL0Hhl/nA6vD4/AH4KLw9LuBr4VfXwvcHX59EfBImsYrVlz3A+dHmT9j+364vxuAh4Fnw+8z\nOl79+YxgDrBNa12ntW4HVgHnZjkmJ+cCD4RfPwB8Nt0daq1fB+qTjONc4EEd4i2gRCk1KoNxxeJc\nYJXWuk1rvR3YRmh7pyOuvVrrteHXTcBGYAxZHrM4ccUiI2MWXu/m8Nv88J8GFgOPhac7x8uM42PA\naUoplcG4YpGxfV8pNRY4E/i/8HtFhserPyeCMcAu2/vdxP+ipBsNvKCUelcpdXV42git9d7w633A\niOyEFjOO3jCG14VPze+zXTrLSlzh0/ATCf2a7DVj5ogLsjxm4csc64ADwIuEzj4atNaBKH1bcYU/\nbwTKMhGX1tqM13+Fx+supVShM64oMfc0PwO+BwTD78vI8Hj150TQ21iotZ4JfBr4ulLqFPuHOnSu\nl3UJV2+JI8yvgeOAGcBe4KfZCkQpNRB4HLhea33U/lk2xyxKXFkfM611h9Z6BjCW0FnHlEzHEA1n\nXEqpauBGQvHNBkqB72cyJqXUWcABrfW7mezXSX9OBB8D42zvx4anZQWt9cfh/weAJwl9Qfab083w\n/wNZCi9WHFkdQ631/vCXNwjcyyeXMjIal1Iqn9DB9vda6yfCk7M+ZtHi6i1jFo6lAXgFmE/o0oo7\nSt9WXOHPBwOHMxTXsvAlNq21bgN+S+bH6yTgHKXUDkKXrxcD/48Mj1d/TgTvAMeH774XELqx8kw2\nAlFKDVBKFZvXwBnA+nA8l4Vnuwx4OhvxxYnjGeDLYQXFPKDRdjkk7TiuyZ5HaMxMXBeFFRQTgeOB\nt9MUgwJWABu11nfaPsrqmMWKK9tjppQappQqCb8uApYQun/xCnB+eDbneJlxPB94OXyGlYm4NtmS\nuSJ0Hd4+XmnfjlrrG7XWY7XWEwgdo17WWn+JTI9XT9xx7q1/hO78byF0jfLfsxhHOSHFxnvABhML\noWt7fwW2Ai8BpRmIZSWhSwZ+Qtcer4wVByHFxC/D4/cBUJPhuH4X7vf98BdglG3+fw/HtRn4dBrj\nWkjoss/7wLrw32eyPWZx4srqmAHTgX+G+18P/Mj2HXib0E3qR4HC8HRP+P228OflGY7r5fB4rQce\n4hNlUcb2fVuMi/hENZTR8ZIniwVBEHKc/nxpSBAEQUgCSQSCIAg5jiQCQRCEHEcSgSAIQo4jiUAQ\nBCHHkUQgCF1EKVWilLo223EIQneRRCAIXaeEkBukIPRpJBEIQte5DTgu7GP//2U7GEHoKvJAmSB0\nkbDr57Na6+oshyII3ULOCARBEHIcSQSCIAg5jiQCQeg6TYTKRApCn0YSgSB0Ea31YeANpdR6uVks\n9GXkZrEgCEKOI2cEgiAIOY4kAkEQhBxHEoEgCEKOI4lAEAQhx5FEIAiCkONIIhAEQchxJBEIgiDk\nOJIIBEEQcpz/H5MLG+1UFXr7AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## Simulation of thebirth death process\n",
    "\n",
    "a = 0.3\n",
    "b = 0.35\n",
    "g = 1-a-b\n",
    "data = []\n",
    "\n",
    "steps = 400\n",
    "\n",
    "for i in range(200):\n",
    "    x = 0\n",
    "    traj = [x]\n",
    "    for j in range(steps):\n",
    "        r = random.random()\n",
    "        if x == 0:\n",
    "            if r < a:\n",
    "                x = x + 1\n",
    "        else:\n",
    "            if r < a:\n",
    "                x = x + 1\n",
    "            elif r < a + b:\n",
    "                x = x - 1\n",
    "        traj.append(x)\n",
    "    data.append(traj)\n",
    "    plt.step(range(steps+1),traj,'lightgray')\n",
    "    \n",
    "data = np.array(data)\n",
    "means = np.array(range(steps),dtype=np.float)\n",
    "for i in range(steps):\n",
    "    means[i] = np.mean(data[:,i])\n",
    "    \n",
    "plt.plot(means,'blue', label=\"Emperical Average\")\n",
    "mu = a / ( b - a )\n",
    "mu2 = a*(a+b) / (a-b)**2\n",
    "std = np.sqrt(mu2 - mu**2)\n",
    "plt.plot([0,steps],[mu,mu],'red', label=\"True stantionary average\");\n",
    "plt.plot([0,steps],[mu-std,mu-std],'pink');\n",
    "plt.plot([0,steps],[mu+std,mu+std],'pink');\n",
    "plt.xlabel(\"t\")\n",
    "plt.ylabel(\"queue size\")\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "heading_collapsed": true
   },
   "source": [
    "## Example: Gambler's Ruin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "Suppose a gambler initially has one dollar. She places a bet against the house that she wins with probability $p$ and loses with probability $1-p$. If she wins, she gets another dollar, otherwise she looses the dollar. She continues this process until she either runs out of money or gets to the house limit of $N$ dollars. For example, if $N=5$, we get the following system:\n",
    "\n",
    "$$\n",
    "  1 \n",
    "  \\hookrightarrow 0\n",
    "  \\mathrel{\\substack{p\\\\\\rightleftarrows\\\\ 1-p}} 1 \n",
    "  \\mathrel{\\substack{p\\\\\\rightleftarrows\\\\ 1-p}} 2\n",
    "  \\mathrel{\\substack{p\\\\\\rightleftarrows\\\\ 1-p}} 3 \n",
    "  \\mathrel{\\substack{p\\\\\\rightleftarrows\\\\ 1-p}} 4\n",
    "  \\mathrel{\\substack{p\\\\\\rightleftarrows\\\\ 1-p}} 5\n",
    "  \\hookleftarrow\n",
    "  1\n",
    "$$\n",
    "\n",
    "**States:**\n",
    "\n",
    "- Absorbing: $0$ and $N$\n",
    "- Transient States: $1$ through $N-1$\n",
    "- Reducibile: No. \n",
    "- Communication Classes: $\\{0\\}$, $\\{1,...,N-1\\}$, $\\{N\\}$\n",
    "- Periodicity: $1$ and $N$ have period $1$. All other states have period $2$. \n",
    "- Ergodic States: $0$ and $N$ are ergodic. \n",
    "\n",
    "**Trajectories:**\n",
    "\n",
    "Once the state becomes $0$ or $5$, the system stays there.\n",
    "\n",
    "**Probability Matrix:**\n",
    "\n",
    "The probability matrix $P \\in \\mathbb{R}^{(N-1) \\times (N-1)}$ with, for example, if $N = 5$, is given by \n",
    "\n",
    "$$\n",
    "P = \n",
    "\\begin{pmatrix}\n",
    "1   & 1-p   & 0   & 0   & 0   & 0     \\\\\n",
    "0   & 0   & 1-p   & 0   & 0   & 0     \\\\\n",
    "0   & p   & 0     & 1-p & 0   & 0     \\\\\n",
    "0   & 0   & p     & 0   & 1-p & 0     \\\\\n",
    "0   & 0   & 0     & p   & 0   & 0     \\\\\n",
    "0   &  0  & 0     & 0   & p   & 1   \n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "**Steady State Behavior:**\n",
    "\n",
    "With this system, the equation\n",
    "\n",
    "$$\n",
    "P x = x\n",
    "$$\n",
    "\n",
    "has two solutions, meaning that $1$ is an eigenvalue with multiplicity 2. The eigenvectors associated with these two solutions are \n",
    "\n",
    "$$\n",
    "u = \\begin{pmatrix}\n",
    "1 \\\\\n",
    "\\vdots \\\\\n",
    "0\n",
    "\\end{pmatrix} \\; \\mathrm{and} \\;\n",
    "v = \\begin{pmatrix}\n",
    "0 \\\\\n",
    "\\vdots \\\\\n",
    "1\n",
    "\\end{pmatrix} .\n",
    "$$\n",
    "\n",
    "Any combination of these two vectors is a steady state solution. If we would like to know what the probability of winning $N$ dollars, however, this analysis will not tell us. This is because the steady state depends on how much money the gambler starts with. If she starts with $N$ dollars, then the probability is $1$. If she starts with $N-1$ dollars, it is more complicated. \n",
    "\n",
    "To determine the steady state starting with $1$ dollar, we can argue as follows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-18T22:43:48.866003Z",
     "start_time": "2019-11-18T22:43:47.871055Z"
    },
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA6UAAAHiCAYAAAAQ42q7AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nA7l8q4Low6iGAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 1152x576 with 8 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#\n",
    "# Gambler's Ruin Simulation\n",
    "#\n",
    "\n",
    "def gamble(p,n):\n",
    "    x = 1\n",
    "    trajectory = [x]\n",
    "    while x != 0 and x != n:\n",
    "        if random.random() < p:\n",
    "            x += 1\n",
    "        else:\n",
    "            x -= 1\n",
    "        trajectory.append(x)\n",
    "    return trajectory\n",
    "\n",
    "p = 0.75\n",
    "n = 10\n",
    "gamble_trajectories,ax = plt.subplots(2,4,figsize=(16,8))\n",
    "\n",
    "for i in range(8):\n",
    "    x = gamble(p,n)\n",
    "    t =range(len(x))\n",
    "    a = ax[int(i/4)][i%4]\n",
    "    a.step(t,x)\n",
    "    if x[-1] == n:\n",
    "        a.set_title(\"Win :-)\")\n",
    "    else:\n",
    "        a.set_title(\"Ruin :-(\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-18T22:40:07.250354Z",
     "start_time": "2019-11-18T22:40:06.970116Z"
    },
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(matrix([[0.65738116],\n",
       "         [0.001121  ],\n",
       "         [0.        ],\n",
       "         [0.00440034],\n",
       "         [0.        ],\n",
       "         [0.00815309],\n",
       "         [0.        ],\n",
       "         [0.00988718],\n",
       "         [0.        ],\n",
       "         [0.00566244],\n",
       "         [0.31339479]]), 0.3392158552348125)"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA7AAAADxCAYAAADyUVuWAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAACDoMACAAAwCAosAAAAg/B/ASongGMzzjCNAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 1152x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 10\n",
    "prob = 0.6\n",
    "\n",
    "P = np.zeros((n+1,n+1))\n",
    "P[0,0] = 1\n",
    "P[n,n] = 1\n",
    "for i in range(n-1):\n",
    "    P[i+2,i+1] = prob\n",
    "    P[i,i+1] = 1-prob\n",
    "\n",
    "p = np.zeros(n+1)\n",
    "p[1] = 1\n",
    "\n",
    "trajectory = [p]\n",
    "for t in range(50):\n",
    "    p = P.dot(p)\n",
    "    trajectory.append(p)\n",
    "\n",
    "trajectory = np.matrix(trajectory).transpose()\n",
    "    \n",
    "fig,ax = plt.subplots(1,1,figsize=(16,5))\n",
    "ax.imshow(trajectory,cmap=\"gray_r\")\n",
    "ax.set_xlabel(\"t\")\n",
    "ax.set_ylabel(\"amount\");\n",
    "\n",
    "q = (1-prob)/prob\n",
    "trajectory[:,-1], (1-q) / (1-q**n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-18T22:42:39.134024Z",
     "start_time": "2019-11-18T22:42:39.112864Z"
    },
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & p \\left(- p + 1\\right)^{2} - p + 1 & \\left(- p + 1\\right)^{2} & \\left(- p + 1\\right)^{3} & 0\\\\0 & 0 & 2 p \\left(- p + 1\\right)^{2} & 0 & 0\\\\0 & 2 p^{2} \\left(- p + 1\\right) & 0 & 2 p \\left(- p + 1\\right)^{2} & 0\\\\0 & 0 & 2 p^{2} \\left(- p + 1\\right) & 0 & 0\\\\0 & p^{3} & p^{2} & p^{2} \\left(- p + 1\\right) + p & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡             2                    2               3      ⎤\n",
       "⎢1  p⋅(-p + 1)  - p + 1    (-p + 1)        (-p + 1)      0⎥\n",
       "⎢                                                         ⎥\n",
       "⎢                                    2                    ⎥\n",
       "⎢0           0           2⋅p⋅(-p + 1)          0         0⎥\n",
       "⎢                                                         ⎥\n",
       "⎢         2                                          2    ⎥\n",
       "⎢0     2⋅p ⋅(-p + 1)           0         2⋅p⋅(-p + 1)    0⎥\n",
       "⎢                                                         ⎥\n",
       "⎢                           2                             ⎥\n",
       "⎢0           0           2⋅p ⋅(-p + 1)         0         0⎥\n",
       "⎢                                                         ⎥\n",
       "⎢            3                 2         2                ⎥\n",
       "⎣0          p                 p         p ⋅(-p + 1) + p  1⎦"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = sm.symbols(\"p\")\n",
    "P = sm.Matrix([\n",
    "    [1, 1-p, 0,   0,   0],\n",
    "    [0, 0,   1-p, 0,   0],\n",
    "    [0, p,   0,   1-p, 0],\n",
    "    [0, 0,   p,   0,   0],\n",
    "    [0, 0,   0,   p,   1]\n",
    "])\n",
    "P*P*P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-18T22:43:30.993783Z",
     "start_time": "2019-11-18T22:43:30.877124Z"
    },
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}-1 & 1 & 0 & \\frac{\\sqrt{2} + 2}{\\sqrt{2} + 2} & \\frac{- \\sqrt{2} + 2}{- \\sqrt{2} + 2}\\\\2 & 0 & 0 & -2 - \\sqrt{2} & -2 + \\sqrt{2}\\\\0 & 0 & 0 & \\sqrt{2} \\left(\\sqrt{2} + 2\\right) & - \\sqrt{2} \\left(- \\sqrt{2} + 2\\right)\\\\-2 & 0 & 0 & -2 - \\sqrt{2} & -2 + \\sqrt{2}\\\\1 & 0 & 1 & 1 & 1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0 & 0\\\\0 & 0 & 1 & 0 & 0\\\\0 & 0 & 0 & - \\frac{\\sqrt{2}}{2} & 0\\\\0 & 0 & 0 & 0 & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "⎛⎡            √2 + 2       -√2 + 2    ⎤                     ⎞\n",
       "⎜⎢-1  1  0  ──────────   ──────────── ⎥, ⎡0  0  0   0    0 ⎤⎟\n",
       "⎜⎢            ⎛√2    ⎞     ⎛  √2    ⎞ ⎥  ⎢                 ⎥⎟\n",
       "⎜⎢          2⋅⎜── + 1⎟   2⋅⎜- ── + 1⎟ ⎥  ⎢0  1  0   0    0 ⎥⎟\n",
       "⎜⎢            ⎝2     ⎠     ⎝  2     ⎠ ⎥  ⎢                 ⎥⎟\n",
       "⎜⎢                                    ⎥  ⎢0  0  1   0    0 ⎥⎟\n",
       "⎜⎢2   0  0    -2 - √2       -2 + √2   ⎥  ⎢                 ⎥⎟\n",
       "⎜⎢                                    ⎥  ⎢         -√2     ⎥⎟\n",
       "⎜⎢0   0  0  √2⋅(√2 + 2)  -√2⋅(-√2 + 2)⎥  ⎢0  0  0  ────  0 ⎥⎟\n",
       "⎜⎢                                    ⎥  ⎢          2      ⎥⎟\n",
       "⎜⎢-2  0  0    -2 - √2       -2 + √2   ⎥  ⎢                 ⎥⎟\n",
       "⎜⎢                                    ⎥  ⎢               √2⎥⎟\n",
       "⎜⎣1   0  1       1             1      ⎦  ⎢0  0  0   0    ──⎥⎟\n",
       "⎝                                        ⎣               2 ⎦⎠"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.subs(p,sm.Rational(1,2)).diagonalize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Page Rank Algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a set $V$ of web pages that all match a given search term. If there are millions of such pages, the question is: in which order should you present them to the user?\n",
    "\n",
    "One possibility is to choose the page with the most incoming links. However, it would seem that links from pages with more links should be weighted more.\n",
    "\n",
    "The solution of Page and Brin treats this problem as a stochastic process. The idea is that a web server starts at some page and randomly follows links (forever). The stationary probability distribution of this process would define the probability that the surfer was at any given page.\n",
    "\n",
    "To make this concrete, we define \n",
    "\n",
    "$$\n",
    "Q(u,v) = \\frac{1}{d^{+}(u)}\n",
    "$$\n",
    "\n",
    "where $u,v \\in V$ and $d^+(u)$ is the *out-degree* of $u$ (that is, the number of pages linking from page $u$ to some other page). 1) If a dead end page $v$ does not link to any other pages, we set its entire column to $1/n$, where $n = |V|$. \n",
    "\n",
    "This version of the Page Rank Algorithm works well, but has one problem. If the surfer encounters a dead end, then she stays there forever. To fix this, Page and Brin added the idea that the surfer would randomly move to an arbitrary new page with some probability $q$, called the \"exploration rate\". The default is $q = 0.15$. \n",
    "\n",
    "The matrix that models the surfer moving completely randomly has all entries equal to $\\frac{1}{n}$, and can be written $\\frac{1}{n} \\mathbb{1}_{n \\times n}$. Here, $n = |V|$, the number of pages. The modified matrix for the Page Rank Algorithm is thus\n",
    "\n",
    "$$\n",
    "\\tilde Q = (1-q) Q + \\frac{q}{n} \\mathbb{1}_{n \\times n}.\n",
    "$$\n",
    "\n",
    "This modification of the transition matrix has the effect of randomly mixing the states and also makes all states ergodic. Thus, $\\tilde Q$ will have a single eigenvalue equal to 1 and a single stationary distribution, in which the probability of a node corresponds to the \"importance\" of the page, and therefore its rank.\n",
    "\n",
    "**Example:** Consider the network of pages in (a) below. The Markov process $Q$ associated with this network of pages in shown in (b). The modified network with an exploration rate of $q$ is shown in (c). \n",
    "\n",
    "<img src=\"https://raw.githubusercontent.com/klavins/EE502P/master/images/page-rank\n",
    ".jpeg\" width=80%>\n",
    "\n",
    "As a matrix, we have\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\tilde Q & = & (1-q) \\begin{pmatrix}\n",
    "0 & \\frac{1}{2} & \\frac{1}{2} & 0 \\\\\n",
    "0 & 0 & 1 & 0 \\\\\n",
    "0 & 0 & 0 & 1 \\\\\n",
    "\\frac{1}{4} & \\frac{1}{4} & \\frac{1}{4} & \\frac{1}{4}\n",
    "\\end{pmatrix} + \\frac{q}{4} \\begin{pmatrix}\n",
    "1 & 1 & 1 & 1 \\\\\n",
    "1 & 1 & 1 & 1 \\\\\n",
    "1 & 1 & 1 & 1 \\\\\n",
    "1 & 1 & 1 & 1 \n",
    "\\end{pmatrix} \\\\\n",
    "& = & \\begin{pmatrix}\n",
    "\\frac{q}{4} & \\frac{1}{2} - \\frac{q}{4} & \\frac{1}{2} -\\frac{q}{4} & \\frac{q}{4} \\\\\n",
    "\\frac{q}{4} & \\frac{q}{4} & 1 - \\frac{3q}{4} & \\frac{q}{4} \\\\\n",
    "\\frac{q}{4} & \\frac{q}{4} & \\frac{q}{4} & 1 - \\frac{3q}{4} \\\\\n",
    "\\frac{1}{4} & \\frac{1}{4} & \\frac{1}{4} & \\frac{1}{4}\n",
    "\\end{pmatrix}\n",
    "\\end{eqnarray}\n",
    "\n",
    "Substituting in $q = 0.15$, the left eigenvector associated with the eigenvalue 1 is of $\\tilde Q$ are found to be\n",
    "\n",
    "$$\n",
    "(\n",
    "0.120452 \\;\\; \n",
    "0.17164409  \\;\\; \n",
    "0.31754157 \\;\\; \n",
    "0.39036233\n",
    ") .\n",
    "$$\n",
    "\n",
    "Thus, the ranking of the pages is: 4,3,2,1. \n",
    "\n",
    "Below, we show two different ways to find this ranking with Python."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 529,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T23:59:29.359636Z",
     "start_time": "2019-11-21T23:59:29.343323Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{q}{4} & - \\frac{q}{4} + \\frac{1}{2} & - \\frac{q}{4} + \\frac{1}{2} & \\frac{q}{4}\\\\\\frac{q}{4} & \\frac{q}{4} & - \\frac{3 q}{4} + 1 & \\frac{q}{4}\\\\\\frac{q}{4} & \\frac{q}{4} & \\frac{q}{4} & - \\frac{3 q}{4} + 1\\\\\\frac{1}{4} & \\frac{1}{4} & \\frac{1}{4} & \\frac{1}{4}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ q     q   1     q   1       q    ⎤\n",
       "⎢ ─   - ─ + ─   - ─ + ─       ─    ⎥\n",
       "⎢ 4     4   2     4   2       4    ⎥\n",
       "⎢                                  ⎥\n",
       "⎢ q      q       3⋅q          q    ⎥\n",
       "⎢ ─      ─     - ─── + 1      ─    ⎥\n",
       "⎢ 4      4        4           4    ⎥\n",
       "⎢                                  ⎥\n",
       "⎢ q      q         q        3⋅q    ⎥\n",
       "⎢ ─      ─         ─      - ─── + 1⎥\n",
       "⎢ 4      4         4         4     ⎥\n",
       "⎢                                  ⎥\n",
       "⎣1/4    1/4       1/4        1/4   ⎦"
      ]
     },
     "execution_count": 529,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Symbolically build the transition matrix\n",
    "\n",
    "onefourth = sm.Rational(1,4)\n",
    "onehalf = sm.Rational(1,2)\n",
    "\n",
    "q = sm.symbols(\"q\",positive=True)\n",
    "Q = sm.Matrix([\n",
    "  [0,onehalf,onehalf,0],\n",
    "  [0,0,1,0],\n",
    "  [0,0,0,1],\n",
    "  [onefourth,onefourth,onefourth,onefourth]\n",
    "])\n",
    "D = sm.Matrix([\n",
    "    [1,1,1,1],\n",
    "    [1,1,1,1],\n",
    "    [1,1,1,1],\n",
    "    [1,1,1,1]\n",
    "])\n",
    "Qt = (1-q)*Q + (q/4)*D\n",
    "Qt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 530,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T23:59:30.164888Z",
     "start_time": "2019-11-21T23:59:30.155602Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.0375 0.4625 0.4625 0.0375]\n",
      " [0.0375 0.0375 0.8875 0.0375]\n",
      " [0.0375 0.0375 0.0375 0.8875]\n",
      " [0.25   0.25   0.25   0.25  ]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([0.120452  -0.j, 0.17164409-0.j, 0.31754157-0.j, 0.39036233-0.j])"
      ]
     },
     "execution_count": 530,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Convert it to numpy and find the eigenvalues\n",
    "\n",
    "Qtn = np.array(Qt.subs(q,0.15)).astype(np.float64)\n",
    "print(Qtn)\n",
    "v = np.linalg.eig(np.transpose(Qtn))[1][:,0]\n",
    "v = v / sum(v)\n",
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 531,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T23:59:31.146390Z",
     "start_time": "2019-11-21T23:59:30.985101Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAb4AAAEuCAYAAADx63eqAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nHxERyQoHHxERycr/AVi8/K62NlwDAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Alternatively, we build a graph and have networkx do the work\n",
    "G = nx.DiGraph()\n",
    "G.add_nodes_from([0,1,2,3])\n",
    "G.add_edges_from([(0,1),(0,2),(1,2),(2,3)])\n",
    "nx.draw(G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 533,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-21T23:59:54.144656Z",
     "start_time": "2019-11-21T23:59:54.139916Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[0.0375, 0.4625, 0.4625, 0.0375],\n",
       "        [0.0375, 0.0375, 0.8875, 0.0375],\n",
       "        [0.0375, 0.0375, 0.0375, 0.8875],\n",
       "        [0.25  , 0.25  , 0.25  , 0.25  ]])"
      ]
     },
     "execution_count": 533,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Here we check to make sure networkx is giving us the same matrix\n",
    "nx.google_matrix(G) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 534,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-22T00:00:02.684570Z",
     "start_time": "2019-11-22T00:00:02.679903Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dict_values([0.12045209234069676, 0.17164393520344706, 0.31754105675824906, 0.39036291569760717])"
      ]
     },
     "execution_count": 534,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now we compute the page rank vector\n",
    "nx.pagerank(G).values()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 632,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-22T01:23:14.682538Z",
     "start_time": "2019-11-22T01:23:12.985897Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0455648506306705 2\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjwAAAIuCAYAAAC7EdIKAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4zLCBo\ndHRwOi8vbWF0cGxvdGxpYi5vcmcvnQurowAAIABJREFUeJzs3XlQlGeeB/BvHxxNN3dzdyNHc18C\nEsGLKCJjvEWjiZpER5NMUjNbW1MztVuzW5Pa2to/dre2kqnJJDNj5TSJtyhC5EYRD45G5JBTkRs5\nm6Npmu5+9w+r35GIytHdtJ3f55+ZtM37PiD4fnme3/N7OAzDMCCEEEIIsWDcpR4AIYQQQoixUeAh\nhBBCiMWjwEMIIYQQi0eBhxBCCCEWjwIPIYQQQiweBR5CCCGEWDwKPIQQQgixeBR4CCGEEGLxKPAQ\nQgghxOJR4CGEEEKIxaPAQwghhBCLR4GHEEIIIRaPAg8hhBBCLB4FHkIIIYRYPAo8hBBCCLF4FHgI\nIYQQYvEo8BBCCCHE4lHgIYQQQojFo8BDCCGEEItHgYcQQgghFo8CDyGEEEIsHgUeQgghhFg8CjyE\nEEIIsXgUeAghhBBi8SjwEEIIIcTiUeAhhBBCiMWjwEMIIYQQi0eBhxBCCCEWjwIPIYQQQiweBR5C\nCCGEWDwKPIQQQgixeBR4CCGEEGLxKPAQQgghxOJR4CGEEEKIxaPAQwghhBCLR4GHEEIIIRaPAg8h\nhBBCLB4FHkIIIYRYPAo8hBBCCLF4FHgIIYQQYvEo8BBCCCHE4lHgIYQQQojFo8BDCCGEEItHgYcQ\nQgghFo8CDyGEEEIsHgUeQgghhFg8CjyEEEIIsXgUeAghhBBi8SjwEEIIIcTiUeAhhBBCiMWjwEMI\nIYQQi0eBh5Cfofz8fMTExGDFihUoLy9f6uEQQojRcRiGYZZ6EIQQ0+nr60NAQACUSiUAwNHRET09\nPRAIBEs8MkIIMR6a4SHkZ6atrQ18Pp/9b7Vajf7+/iUcESGEGB8FHkJ+ZiIjI2FnZwcejwcrKytI\npVL4+Pgs9bAIIcSo+C9+CyHEkgwMDOCf/umfMD4+joGBAfznf/4neDzeUg+LEEKMimZ4CPmZUKvV\nuHz5Mr7++muo1Wr827/9G7y8vHD9+vWlHhohhBgdzfAQ8jPQ2tqK8+fPY2pqCgzDwM7ODu3t7eBw\nOKipqYFMJkNkZORSD5MQQoyGZngI+Rl48OABpqamoNVqAQB2dna4f/8+GIaBTqfDpUuX0NPTs8Sj\nJIQQ46HAQ8jPwMaNG7FmzRrweDxwuVzY2dmhubmZ/XOdToesrKwlHCEhhBgXLWkR8jMwNTUFuVyO\nQ4cOobu7G9bW1igtLYWnpycePXqEnTt3IjAwcKmHSQghRkOBh5CfgaKiIgQGBmLZsmVYtmwZACA+\nPh4A8PXXX8PW1pYaDxJCLBotaRFi4fr6+lBTU4ONGzfO+ucSiQQdHR0mHhUhhJgWBR5CLBjDMMjK\nysL69eshFApnfY9UKkVnZ6eJR0YIIaZFgYcQC1ZdXQ2tVou4uLhnvkcikaCrqws6nc6EIyOEENOi\nwEOIhZqcnER+fj62bNkCLvfZP+p2dnYQCoV0nhYhxKJR4CHEQhUWFiIsLAze3t4vfK9UKqU6HkKI\nRaPAQ4gF6u7uRkNDAzZs2DCn90skEqrjIS8NhmGeek2n0+Ff/uVfsGfPHvT29i7BqIi5o23phFgY\nfRPBlJSUOW81l0qluHXrlpFHRohhZGRkoKmpCW5ubpBKpXBzc0NXVxc++eQTqNVq+Pj44JNPPlnq\nYRIzQ4GHEAsjl8vB4/EQExMz549xc3PD2NgYlEol7OzsjDg6QhbP3d0dtbW16OjoYJdinZycYGVl\nhenpaToXjsyKAg8hFmRiYgJFRUV46623wOFw5vxxXC6XXdYKDg424ggJWRi1Wo22tjY0NzejoaGB\n3VXI4/EQFhaGkJAQTExMYHx8HNbW1mAYZl4/A8TyUeAhxILk5+cjKioKHh4e8/5YfQNCCjzEXAwO\nDqK5uRktLS3o6OiAt7c3ZDIZDh48iG+++QZTU1NISEiAt7c3Lly4AJ1OB4FAgImJCTQ0NCAsLGyp\nPwViRijwEGIhOjo60Nraig8//HBBHy+RSHDjxg0Dj4qQuZuensbDhw/R3NyM5uZmaDQayGQyxMfH\nY+/evbCxsWHfm5SUxLZbuHTpEnQ6HbhcLkQiEVavXo3c3FwEBQWBz6fHHHmMvhMIsQD6QuXU1NQZ\nD4X5kEgk6O7uZh8chJjC8PAwO4vz8OFDeHp6QiaT4fXXX4eHh8czl6XWrFmDlpYWfPfdd+xrfD4f\nIpEIDg4O8PDwwK1bt7BmzRpTfSrEzFHgIcQClJeXQyAQLKpYUyAQwMHBAX19ffDy8jLg6Aj5B41G\ng/b2djbkTE5OQiaTISYmBrt27ZrXIbb+/v7Ys2cPLl++DLVaDY1GA4FAgJGREWzatAnHjx9HTEwM\n7O3tjfgZkZcFBR5CXnJjY2O4du0a3nnnnUUXaeoLlynwEENSKBRswGlra4ObmxtkMhl27doFLy+v\nBX/f8ng8LFu2DABw9OhRNDU1QaVSYWRkBLGxsYiNjUVhYSF27NhhyE+HvKQo8BDyksvLy0NsbCzc\n3NwWfS2pVIq2tjYkJCQYYGTk50qr1aKjo4MNOePj4wgMDERERAS2b99u0NYH1dXVCA0NhZeXF7y8\nvHDnzh08ePAAALBu3Tr8+c9/Rnd395w6jhPLRoGHkJdYW1sb2tvb8cEHHxjkelKpFNevXzfItcjP\ny9jYGBtw7t+/D1dXV8hkMmzbtg3e3t5GqQtjGAZyuRy7du1iX3NycsLIyAgAwMbGBuvXr0dOTo5B\nZkDJy40CDyEvKa1Wi+zsbKSlpcHa2tog1xSLxZicnMTExASEQqFBrkksk06nQ2dnJxtyRkZGEBgY\niJCQELz22msQiURGH0NbWxv4fD58fHzY154MPACwfPlylJeXo76+HhEREUYfEzFfFHiI0Z04cQIf\nfPABVq9ejczMTNomaiC3bt2Co6MjQkNDDXZNDocDHx8fdHR0GPS6xDKMj4+jpaUFLS0taG1thZOT\nE2QyGTZv3gyJRGLy3X1yuRxxcXEzZm4cHBwwMTEBrVYLHo8HLpeLtLQ0ZGRkIDg4GFZWViYdIzEf\n9OQhRvcf//EfGBsbQ0FBAbKzsxEfH4+pqSkolUpMTk7Cx8eH1tfnaXR0FKWlpTh69KjBp+n1DQgp\n8BCdTofu7m52FmdwcBABAQEICgpCWlraku5+UiqVaG5uxmuvvTbjdX0vntHRUTg7OwMA/Pz84O3t\njZs3b2LdunVLMVxiBijwEKM7evQo/vVf/xVSqRTl5eWoqqoCl8sFn8+HRqPBq6++SoFnnnJycpCQ\nkAAXFxeDX1sqleLatWsGvy55OSiVSrS2tqK5uRmtra0QiUSQyWRITU2FVCoFj8db6iECeFysHBIS\nMus2dv2ylj7wAEBqair+/ve/Y/ny5XBwcDDlUImZoMBDjO73v/893Nzc4O/vj+joaHz22WfQaDRQ\nq9UAQIdVzlNrayt6enqwc+dOo1xfIpGgp6eHXRIglo1hGPT09LCzOP39/fDz84NMJkNKSgocHR2X\neohP0Rcrb926ddY//2kdDwA4OzsjPj4eBQUFM4qcyc8HBR5idNPT0+js7ERXVxdWrlyJw4cP46uv\nvsL09DQEAgGysrKQm5uLqKgobNy4Eba2tks9ZLOl0WiQnZ2NzZs3G60WwcbGBs7Ozujr66OZNws1\nOTmJ+/fvsyFHIBBAJpNh/fr18PX1Nfs6u46ODjAMA19f31n/3MnJCcPDw0+9vmbNGnz66afo7OyE\nRCIx9jCJmTHv72piEerr68HhcKDT6XDlyhXs2rULe/bswQ8//IDXX38dEokExcXFkMvlqKyshLe3\nN1JTU+Hn57fUQzc7paWlcHd3R1BQkFHvo6/jocBjGRiGQV9fHxtwent7sWzZMshkMiQnJ89Y+nkZ\nzFas/CQnJye2F8+TbGxssGHDBuTk5ODIkSO0Tf1nhgIPMbpbt25Bq9UCeBx+4uPjERwcjH/+539m\n19I3btyIjRs3orW1FQUFBfj6669hZ2eH+Ph4rFu3zux/4zSF4eFh3L59G++++67R7yWVStHa2oqV\nK1ca/V7EOKampmbM4vD5fAQFBWHt2rVYtmzZS7tbaXJyEg0NDdi0adMz3zPbkpZeTEwMysvLUVNT\ng+joaGMNk5gheooQoxoZGUFvby+srKyg0Wig1Wohl8vh6+s7a+FgYGAgAgMDoVQqkZ+fj1u3buH6\n9evw8/NDWloaPDw8luCzMA9XrlxBUlISnJycjH4v/awbeXkwDIP+/n424HR3d0MqlUImk2H16tVw\ndXVd6iEaRE1NDWQy2XNr/54XeDgcDtLS0nDu3DmEhoYarIcVMX8UeIhRiUQi7N+/H5OTk7h27Rre\ne++9Of0DY2dnh+3bt2P79u2oqanB1atX8fnnn8PBwQFJSUl45ZVXflYnejc2NmJwcBCvv/66Se7n\n6uoKtVqNsbExOnjRjKnVajx48ADNzc1obm4Gh8NBUFAQkpKS4OfnN6eftbq6OiiVypfiOBGGYVBZ\nWYm0tLTnvu+nvXh+ytfXF76+vigtLcX69euNNVxiZijwEKPi8/kICQnBxMQEcnNzYWNjM+9rREVF\nISoqCiMjI8jJyUF+fj7y8/MREhKCTZs2meUuEkOanp7GlStXsG3bNpPtmuJwOGwdT3h4uEnuSV6M\nYRgMDg6yszidnZ3w8fGBTCbDwYMHIRaL512XkpiYiPHxcUxOTpr9hoGuri5MT0/D39//ue/jcrmw\nt7eHQqF4ZuuGjRs34q9//Svi4uIs/t8Q8hgFHmISAoEAU1NTi9rq7OTkhH379kGn06GsrAw3b97E\nxx9/DFdXVyQnJyMqKsrAozYPJSUl8PHxQUBAgEnvqz85nQLP0pqenkZbWxs7i6PT6SCTyZCQkIDX\nX399Qb9E6PX39yM5ORkqlWpR1zGVFxUrP0m/rPWswOPo6IiEhATk5+cjPT3d0EMlZogCDzEJLpcL\nOzs7TExMLLrpF5fLRWJiIhITE9HT04O8vDxcuHABly9fRmRkJFJTU83+N9W5GhwcREVFBd5//32T\n31sqlaKoqAjA487Od+/eRWJiIhWQm8DQ0BA7i9Pe3g4vLy/IZDLs378f7u7uBtldpNVqceHCBXz0\n0Ue4efMmuru7Z5xJZW6mpqZw7949fPjhh3N6//PqePRWr16NTz/9FB0dHZBKpYYYJjFj9C8XMRmR\nSITx8XGDdjn18vLCW2+9BY1Gg6tXr6KyshJyuRxeXl5ITU194dS3OWMYBtnZ2Vi7du2SdIb18fFB\nb28vNBoN1q1bh3v37uHYsWP485//bPKxLFRpaSn+/d//HRqNBn/4wx9eWPuxVDQaDR4+fMiGnKmp\nKchkMsTGxiI9Pd0oAb6kpARCoRDx8fEYHR1FXV2dWQeempoa+Pv7z/lQUkdHxxcGHmtra6SkpODK\nlStGOaaFmBcKPMRkRCIRJiYmjHJtPp+PlJQUpKSk4P79+8jPz8c333wDgUCA+Ph4JCcnv3QzE/X1\n9RgfH8crr7yyJPe3traGq6srent7MTY2xp6rZI7kcjl4PB6Cg4PZowbu3buHTZs2QalUAgB2796N\nwsJCs9lqPzIywgactrY2eHh4QCaTIT09HZ6enkZ9+HZ3d6OiogLvvfceOBwOIiIi8P333yM1NdVs\nH/pyuRwbNmyY8/udnJxw//79F74vKioKZWVlqK6uxvLlyxczRGLmXq4nAHmpCYVCjI+PG/0+AQEB\nePfdd6FSqZCXl4eysjKUlpZi2bJl2LRpE7y8vIw+hsWamppCbm4udu/evaTHO+gLl0tKSvDJJ5/A\n0dHRLI+cqKiowKNHjwAA7u7u8Pf3R2VlJdv/CXjcvyUzM3PJAo9Wq0V7ezsbciYmJiCTyRAVFYWd\nO3fOeiaUMWg0GmRkZMw4/NPd3R3W1tbo7Ow0y6Wdnp4eKJXKedWxOTs7v3CGB3hcoP+LX/wCp0+f\nRnh4OG1Tt2AUeIjJ6Je0TMXW1hbbtm3Dtm3bUFtbi6tXr+Jvf/sbHBwckJiYiJUrV5rt1varV6/C\n398fy5YtW9JxSCQSNDU1ISkpCSEhIejo6MClS5fM4iyiyclJDAwMoL+/H1qtlg03PT096OnpAYfD\nAZ/Px9TUFIDHM1aenp4mHePo6CgbcB48eACxWAyZTIYdO3bA29t7SWZTCgsL4ebmhsjISPY1/SxP\nbW2tWQaeyspKxMbGzuvndS41PHoSiQT+/v4oKSlBSkrKQodJzBwFHmIyIpFo1vNtTCEyMhKRkZFQ\nKBTIzc1FQUEBCgoKEBQUhLS0NJM085urR48eobq6Gr/61a+WeiiQSqUoKCgAADY43L17F76+voiP\njzf6/RmGwfj4OPr7+9lwo//f6elpiMViuLm5wdnZGQMDA+BwOBAIBNi/fz8AoKCgAJWVlWAYBq6u\nrti+fbtRx6vVatHZ2cmGnNHRUQQGBiIsLAxbt26FUCg06v1f5OHDh6ipqcGvfvWrp8JWZGQkvv76\na6SlpZnVLwJqtRp1dXXz/nmwt7eHUqmERqOZ03J2SkoKPv/8c8TFxb10R22QuaHAQ0xGKBSis7Nz\nScfg6OiIvXv3QqfToby8HDdu3MAnn3wCV1dXrFu3bslbzesLlZOTk+dcnGlMzs7O0Gq1UCgUUCgU\n7OuXL1+Gg4ODwc700ul0UCgU6O/vZ0ONPtjweDw22IjFYoSGhkIsFsPBwYF9aLe1taGxsRERERFY\nuXIlrl27hqamJhw7dgxpaWlYu3YthEIhMjIycPTo0ed26X2R5uZm7N27FwMDA/jf//1fbNmyBS0t\nLWhpacH9+/fh7OwMmUyGLVu2wMfHx2zCg1qtxsWLF7F169ZZP3+xWAyhUIj29nazOseurq7umZ3Z\nn0ffi2d0dPSZW9Of5ODggJUrVyI/Px979+5d6HCJGaPAQ0zG1Etaz8PlcrFy5UqsXLkSfX19yM3N\nRUZGBru1fePGjYt6KC5UTU0N1Go1VqxYYfJ7z4bD4UAqlaKtrQ1TU1PgcrnQ6XTgcrno6uqad+DR\narUYHBx8arZmcHAQdnZ2bKjx8fHB8uXL4ebmNqe/B19fX6Snp+P27dv45ptvoNFowOFw4OzsjNbW\nVnC5XMTFxWFgYABnz57FwYMHFxxEDh06hLt374JhGBw6dAh/+MMfEBERgaCgIGzevNksgupscnNz\nsWzZMoSEhDzzPREREairqzOrwCOXy7FmzZoFfeyLevH81KpVq/Dpp5+ira3NrL4GxDAo8BCTMafA\n8yQPDw8cOnQIGo0G165dQ2VlJaqqquDl5YWUlBQEBgaaZBz6Iut9+/aZzawA8I/CZalUCn9/f9y8\neRPbtm3D1atXsWbNmlmXC9Rq9YxQo///IyMjcHR0ZIONTCZDYmIixGLxohrfcblcODs7s9vogcdh\n7c6dOwAeL8OtXr0aGzduxA8//ICcnBxs3rx53vdRKBTo7OwEwzAAHs/ILV++HGlpaSYrOl4I/QzU\ni/o5RURE4IsvvsDmzZvN4nuwr68PCoViwTOJ86njAQArKyts3LgROTk5OHbsmFl8DYjhUOAhJmPM\nbemGwOfzsWHDBmzYsAFtbW3Iz8/HiRMnIBAIEBsbi/Xr1xt1a3tRURGCg4MhkUiMdo+FkEqlaGho\nwNGjRwEAvb290Ol0cHFxwbVr1yCTyZ6qsZmYmICrqyvEYjHEYjEiIyPh5uYGFxcXo30NfXx8sHXr\nVly8eJFd6tIXMg8NDWFkZAROTk5IT0/H8ePH2a69czE2Nobr16+jpqYGv/nNb/DRRx9henoaiYmJ\nmJycxJ/+9CeEhYUhISHB7HYB6nen7dix44X9fFxcXODg4IC2tjaTd/aejVwun3ex8pPmG3iAx6Gv\nvLwcd+7cmfP3B3k58D766KOPlnoQ5OeBz+ejqKgIq1evNvvfnJycnBAXF8eeM1RVVYWrV6/iwYMH\n8PDwMPiBmj09PSgoKMC+fftgZWVl0Gsvlp2dHXJzc+Hl5YWWlhb09fWhuroao6OjaG1tRU9PDxiG\ngZOTE4KCgrBq1SqkpqYiISEB4eHh8PPzg7u7O4RCoVH/3kdGRnDu3Dns2rULIyMjUCgU4PP5YBgG\nDMPAzs4Oy5YtA5/PR0BAAM6fPw+pVPrcc5SUSiWKi4tx6dIleHp6Ij09HXw+H8HBwTh58iSSkpJQ\nX1+PgwcPQqlUIjc3F/X19bCysoJYLDaL7/PMzEx4enrOuZ+TSqXCw4cPn7v0ZQrT09NszdFCGy8q\nFAr09PQgLCxszh/D4XDg4eGBzMxMxMfHv3T9u8izUeAhJsPhcCCXyxEREfHSHP2gf7itXbsWbm5u\naGpqQklJCeRyOXQ6HSQSyaK3FjMMg1OnTmHVqlXw9fU10MjnT6fTYXh4GO3t7WhsbERVVRWuX7+O\ngoICaLVaPHr0CDweD97e3rh//z6OHTsGgUAAnU6H9PR0+Pr6QiwWQyAQmHy79fT0NE6cOIEVK1Yg\nNjYWcXFxWL58ObRaLWxsbBATEwNvb2+2lsPOzg7u7u64cOHCrN+PKpUKJSUlyMjIgIuLC/bs2YOI\niAjU19cjJycHGo0GaWlp8PLyglqtRklJCbZu3YpVq1ZBKBSisrISBQUFUKlUcHV1XbJzqurr61Fd\nXY19+/bNuXeSg4MDcnJykJiYuKSBrba2FiqValGNN/XHUcTGxs7r4+zt7dHf34/u7m6TLWkT46Po\nSkxKv6xlTtvA5yo8PBzh4eEYHR1Fbm4uioqKUFhYyG5tX+hW1qqqKnA4nHn/o7xQGo0GQ0NDT+2I\nGhwchFAoZHdESSQSxMbGQiwWo7i4GI6Ojli9ejUAoL29Hb29vVi7di3+8pe/oLm52WA7tuaLYRhk\nZmbCzc0NiYmJ7OtOTk7gcDiQyWRISkp66uOCgoKQlJSEkydP4siRI7CysoJarcbt27dx69YtBAcH\n49ixY+zfq1wux48//giGYcDhcKBSqWBra4u1a9dibGwMp06dwoEDBxAaGorQ0FD09/ejvLwcn332\nGfz9/ZGQkAA/Pz+ThcHx8XFkZ2dj//7985o1dHJygqurK+7fv79kf6fA46/3k3+fC7GQJS29DRs2\n4LPPPkN8fPyci56JeaPAQ0zKVN2WjcnBwQF79uyBTqdDZWUlSktL8ac//QkuLi5Yu3btjPb0Op0O\njx49embDO6VSicLCQhw4cMDgD8KpqakZBcP6/1UoFHBycmILh/XLUGKx+JldZqVSKerr69n/DgsL\nQ319PSIjI7F582b8+OOP8Pf3X5Lp/1u3bmFgYACHDx9+6ms4MjLy3OaNSUlJ6Ovrw4ULFyCVSlFa\nWgp/f38cPnwYYrGYfV97ezsyMzPZ/+bz+RgeHoaXlxc4HA42b96MM2fOICMjA7t37waHw4Gbmxte\ne+01pKSk4O7du/jxxx8BAAkJCYiOjjbqrA/DMLh8+TJiY2MXVBOm3621VIFnYGAAQ0NDCA4OXtR1\n5tuL56cfm5SUxG4kIC8/CjzEpMx1p9ZCcLlcJCQkICEhAY8ePUJubi4yMzORnZ2NiIgIpKamorW1\nFefPn8e+ffsQGhr61DUKCgoQHh6+qEJXpVI5a2M+pVIJV1dXNthER0dDLBbD1dV13kdDSKVS5Obm\nsrMboaGhyMnJgVqtRlBQEORyOUpLS5GcnLzgz2Mh7t+/jxs3buDo0aOzzmLoC5WfRafTwcvLC3l5\neejv78ehQ4fg4eHx1Pu8vb2xfft2FBQUsA9QfeABHn8v7N69GydOnEBubu6MQ0ptbGyQkJCAFStW\n4OHDhygrK0NhYSGioqKQkJAANzc3A3wlZqqursbIyAj27NmzoI8PDw/H1atXFxQUDKGyshIxMTGL\nPsKEy+XCwcEBCoUCrq6u8/74pKQkfPrpp3jw4MFLfRAxeYwCDzEpS5jhmY27uzsOHjwIjUaD69ev\no6KiAnfu3GH/wT5//jzefvvtGadRd3Z2oqmpCR9++OELr88wDMbGxmZtzKfVatlQIxaLERAQADc3\nNzg6OhqsBkNf2KufHRIIBJBIJGhpaUF4eDjS0tLwt7/9DdHR0SbrUjs8PIzz589jz549zyw8flbg\n0el0uHv3Lq5evQpXV1fs27cPly9fxsjIyKyBh8/nIyYmBgUFBfjlL3+J3t5eeHt7z3iPlZUV9u/f\njy+//BI3btzAqlWrZvw5h8OBn58f/Pz8MDo6ioqKCnz99ddwd3dHQkICQkJCDPL3pVAokJeXh0OH\nDi04rDg4OMDd3R2tra0mL17WaDS4e/cufvnLXxrkevplrYUEHj6fj9TUVFy5cgXvvfeeWRShk4Wj\nwENMSiQSYWBgYKmHYTR8Ph+vvvoqXn31Vdy5cwcXL14E8Lio9ssvv8SxY8eg0WiQn5+Phw8fPrVV\nWF84/NPZmoGBAVhZWbHBxt3dHeHh4XBzc4NIJDJ6XYi+AWFHRwcbIPTLWuHh4XByckJSUhKuXLmC\nN954w6hjAR73+Tl58iTWrl37zAZxKpUKDMPM+PoyDIO6ujoUFxdDJBJh586d7JLX66+/jh9++AHv\nvPPOrLMuDx48gKOjI3x8fGYE1ycJBAIcOHAAX3zxBezt7REVFTXr+xwcHLBhwwasW7cO9+7dw82b\nN3HlyhWsWLECcXFxCz6CgmEYXLx4EYmJiYs+N0y/rGXqwNPQ0AAPDw+D1c04OjouuI4HePx9XlZW\nBrlcbjYNQcnCUOAhJiUSifDw4cOlHoZJdHd3g8vlgs/nQ6PRQKvV4v/+7//w2WefYXp6GjqdDhER\nETMCztDQEEQiETtb4+vri7i4OLi5uS15Yzt9A0L9Qzw0NBR5eXmYnp6GlZUVkpKS8Pnnn6OxsdGo\nD0mGYXDp0iV4e3s/dwePfnaHw+GAYRg0NjaiqKgIVlZW2Lx5MwICAmYERYlEgk2bNuGHH35gd6A9\nqba2dsaBm8/i6OiIAwcO4Ouvv4ZQKHxuPxs+n4+oqChERUWhp6cH5eXl+POf/4zg4GAkJCTAx8dn\nXmG2vLwcarWaLS5fjPDwcBQtI4y5AAAgAElEQVQWFrJ/v6Yyn/5Ic7GYwmXgH6epnzhxApGRkS/N\nDlPyNAo8xKQsqYbnRVasWIGAgAAIBALY2dlBIBAgKysLGo0GKpUKfD4f33//PXbv3g0/P78XFg4v\nNYlEgtraWva/hUIhvL290draitDQUPD5fGzevBmXL19GQECA0R6SpaWlGB4enrVI+Un6rs4tLS0o\nKiqCVqvFhg0bEBwc/MyPi4mJQV9fH86ePYsDBw6wSxgajQaNjY1Yv379nMbo7u6OvXv34syZMzh4\n8OCcarS8vLywfft2pKamoqqqCufPn4etrS0SEhIQGRn5wq/n4OAgiouLceTIEYMsvYhEInh7e6O5\nuRnh4eGLvt5cDA0Noa+vb9Z6t4VycnJCa2vroq7h6emJkJAQXL16dUZ9Fnm50IIkMSlz77ZsSO7u\n7ggNDcWyZcvYpadXX30VPB4PdnZ2sLa2xuuvvw5bW1tUVFTg22+/xZkzZ3D16lXcv3+fPZ3cXHh7\ne2NgYABqtZp9Tb+spRcYGAhvb29cv37dKGNoaWnB7du3sW/fvhfWp9y/fx89PT3IycnBqlWr8N57\n7yEkJOSFMyYbN24Eh8NBbm4u+1prayvc3d3ndYCln58ftmzZgh9++AHDw8Nz/jiBQIBVq1bh17/+\nNdavX4979+7h448/Rl5e3jOvo9PpkJGRgeTk5Bm7yxZLv6xlKnK5HDExMQYtlF7sDI/ehg0bUF1d\nbdFL8paOZniISVlq0fJcMAyDmzdv4pNPPoGDgwPi4+Nn7PwYGxtDZ2cnOjo6UFxcjN7eXri4uEAi\nkUAqlUIikcDFxcXkTf30+Hw+3N3d0d3dzdbNhIWFobCwcMZunrS0NHz++eeIiYkxaP+SoaEhXLhw\nAfv27Xtu8Ojs7ERRURG6urogk8mwe/fuec14cLlc9vgJDw8PxMbGora2FhEREfMec3h4OMbHx/Hd\nd9/hyJEj8zqQlsPhICgoCEFBQRgaGkJFRQX+/ve/QyqVIiEhAYGBgez3wo0bN8Dn8xfVpG82YWFh\nyMvLg1qtNvrMo1arxZ07d/DOO+8Y9LqGCjxCoRBr1qxBbm4u3nzzTQOMjJgadVomJsXj8XDt2jUk\nJiYuesvpy0Yul6OhoQHvvPMOoqKintrNZGNjAzc3NwQGBiI2NharVq2Ct7c3NBoN2traUFJSguvX\nr6O9vR0jIyPscQmm/Do+evQIKpWK7QhtbW2NlpYWiEQidheMvr9MRUUFoqKiDBLQpqam8O2332LN\nmjXPPCagt7cXmZmZKCsrw4oVK6DT6RAZGQl3d/d538/KygqBgYE4f/48vLy82E7KC3no+/j4YGxs\nDCUlJYiMjFzQ35dAIEBgYCBeeeUVaLVaXL9+HTdv3oROpwPDMLhy5QrefPNNg9d5WVlZoa2tDVZW\nVgv6Os5HY2MjhoeHn9rdtljW1tYoLCzEqlWrFr3U5+Xlxe7so2aELx+a4SEmxeFw2GUtc61VMYbu\n7m4UFhbi8OHDc/68eTweJBLJjMZxo6Oj6OjoQGdnJ/Lz8/Ho0SOIxWL2fVKplC3UNQapVIqampoZ\nr+mXtZ5sEpeYmIg7d+6goaFhXucYzYZhGGRkZEAikSA+Pv6pP+/v70dxcTHa29uxZs0a7N27F3w+\nHxUVFYvq6C0Wi7Fz506cOXMGHh4eC945BTxeDrl48SLOnj2L/fv3L/jBa2Vlhbi4OMTGxqKzsxO3\nb99GXl4efH19oVKpFjy+59Eva82lYHsxKisrjXJY52J78TyJz+dj06ZNyMnJQUBAAG1Tf8lQ4CEm\np1/WMlW/lqWmVCpx+vRpbNmyZdH1FQ4ODoiIiGCXVzQaDXp6etDR0YGGhgbk5eWBYRh2CUwqlcLL\ny8tgBcRSqRTZ2dlsA0LgceApLi6GVqtlZy94PB62bNmCjIwMBAYGLirclpSUYHx8HOnp6TOC3NDQ\nEK5evYqWlhasWrUKO3bsmHGfFzUdnIugoCDY29tDoVAsarcSh8PBtm3bcPLkSWRmZmL79u2LCqX6\nNgHNzc3w9/eHr68vvv/+ezg7OyMhIQFhYWEGm/kLDQ3FlStXMDU1ZbTu0CMjI+ju7jZaR2MnJycM\nDw8vOvAAQEhICMrKylBRUWHwJURiXBR4iMn9nHZqMQyDCxcuICwszCg7Xfh8PqRSKaRSKXs/hULB\n1gLl5OSgv78fbm5uM0KQg4PDgh64Dg4O7LEK+il9BwcHuLm5PXX2kp+fH3x9fVFSUoKUlJQFfX6N\njY2oqKjAsWPH2BohhUKBa9eu4d69e3jllVfwm9/85qkHsUqlgk6nW/QSj0qlwujoKAIDA3Hp0iX2\n2IiF4PF42Lt3L77++msUFRVhw4YNixpbZ2cn5HI53n//fYhEIqxZswaNjY0oLy9HTk4O4uLisGLF\nCtjb2y/qPgKBAH5+fmhoaEBMTMyirvUsVVVViIqKMtrOPicnJygUCoNci8PhIC0tDd988w2ioqKW\nvF0EmTuq4SEm9+DBA1hbWz+zeZsluXbtGgYGBrBr1y6TTH9zOBzY2trC3d0dQUFBiI+PR2JiItzd\n3aFSqdDS0oLi4mLcvn0bnZ2dGB0dBfB41m2u4+vs7ISVldWMxnYqlQptbW1PbSeWSCTIzMxESEjI\nvAp2gcfnKZ06dQr79++Hm5sbxsfHUVBQgKysLPj6+iI9PR1BQUGz7ugZGBhAW1sbEhIS5nXPn6qt\nrYVGo8Hu3btx8+ZNTE5OLupEex6Ph9DQUOTm5rInzy/E9PQ0vvvuO6SmprJLnlwuF25ubli+fDlk\nMhkePnyIrKws9PT0QCQSwdHRccFhTd+w8VmNFBdDv8PsF7/4xaKWDZ+nr68PSqXyuT2R5kMkEmFk\nZAQPHz5c0gNWyfxQ4CEm193djenp6Wd2yLUU+nBx6NChJW1WxuPx4OTkBF9fX0RGRiIpKQmhoaGw\ntrZGX18fKisrkZ+fj6amJvT390OtVsPW1vaZyxdjY2Po6+ubUbPj4OCAnJwcJCYmzghONjY24PF4\nuH37NqKjo+f8wFWpVPjmm2+QnJwMqVSK4uJiXLp0CZ6entizZw9CQ0OfOxvQ2dmJkZGRRT+gCwoK\nEB0dDS8vLwQFBSEzMxMuLi6LWpq0trZGUFAQMjIy4OzsvKCztPLy8mBjY/PMs8uEQiHbvFClUqGo\nqAhyuRwcDgdisXjey11OTk7IyclBfHy8wWdhmpqa8OjRI6xdu9ag132SQqFAd3e3QWdZJRIJsrKy\nEBwcbLSgRgyLAg8xuYGBAQwPDy/6JGRzNjIygu+//x579uwx+u6W+eJwOBAIBPDw8EBwcDBWrFiB\nlStXQiwWY3JyEk1NTSgsLER5eTk6OzsxNjYGDoczYxaooqJiRpt9W1tb3Lt3D87Ozk/tXvH29sbN\nmzdhZ2c3p68FwzA4e/YsGyoyMjLg4uKC9PR0REZGzqkeqLW1FTqdblG/fU9MTCAvLw9bt24Fj8eD\njY0NpFIpzp07t+iHnH6Z6Ny5c5BKpc88C2w2Dx48QElJCd58880Xhg8+nw8fHx8kJCTA1dUVtbW1\nyMnJwdjYGJydnec868bn89HV1QWGYRZ10O1s8vLyEBUVZfDrPmlqagr19fUGLYq2srICj8dDeXm5\nwXYjEuOiGh5iciKRaNGdT82ZRqPBmTNnsGrVKvacJnNnbW0Nf39/ti8QwzAYGhpCR0cHOjo6IJfL\n2dPBfXx8MDAwgKGhoRnhRr9bKzAwcMa1uVwuXnvtNZw/fx4ymeyFha8FBQXo6+vD9PQ0goKCcOzY\nsXkXuBuiYPnevXsICgqaEbAkEglSU1Nx8uRJHD16dFH1G97e3ti9ezdOnz6Nt99+e04zPVNTU7h4\n8SK2bds2r3tzOBwEBAQgICAACoUCFRUV+PLLL+Hp6YmEhAQEBQW9cEkzIiICd+7cQWxs7Jzv+yKj\no6Nob29Henq6wa45G0P14vmphIQEVFRUoLm52aJ/gbMUNMNDTE6lUqGhocEoW1DNQXZ2Nng8HtLS\n0l7a3/o4HA7s7OzYlvoJCQlISEiAi4sLlEolenp6UFFRgaqqKnR1dWFsbAwuLi64fv06kpKSnvq8\nnZyc0NfXh+7u7qcCkZ5Go0FmZiYqKiqwbNkypKenIy4ubkGhorKyku1wvVC5ubmIi4t7avnK09MT\nIyMjBvnN3sXFBSKRCJcuXUJ4ePgLw2B2djacnZ0X1avG1tYWAQEBeOWVV8DhcHDz5k2UlJRAo9FA\nLBY/c9ZIv6wVGxtrsJYSt27dgpOT06JbF7yIlZUVioqKDNKL50lcLhfOzs7Izc1FfHw8bVM3cxR4\niMlptVpUVFQgMTFxqYdicNXV1aiurp7TcsPLhs/nw8XFBX5+flAqlfD398eGDRvA5XLR1dWFO3fu\nYGxsDA0NDVAoFNBoNBAIBOzXQSKR4PLlywgKCpqxHKTVaiGXy3Hy5El0d3djx44dSElJmXeR85NK\nS0sRExOz4B1Ko6OjKC4uxtatW2d9iAUEBKC2thY9PT2QyWQLHicAeHh4QKvVIj8/H1FRUc88VqGp\nqQnl5eXYv3+/QY5e4HK58PT0RFxcHHx9fdHS0oLs7GwMDAzA3t7+qW7WPB4Pvb290Gg0Cy62fpJO\np8PFixeRmpq66J1kL8LhcHDnzh0EBQUt6vtqNq6urmhuboZSqYROp0NNTc1LM7P7c0OBh5gcj8dD\ncXEx1q5d+9LOgMymr68P58+fx4EDB+ZVk/EyUqvVaGpqQmJiIry8vBAaGsp2AZ6enoaDgwPu3buH\n/Px8VFVVoaenB2q1Gs7OzqioqMDy5cvBMAyqq6tx5swZjI2NYWpqCmlpaQbZ+lxQUIDk5OQFh065\nXA6BQPDMIlf9sQ/5+fmwtrZedP2JVCrF4OAgbt26hcjIyKdCllKpxPfff49du3YZ9KwsPXt7e4SG\nhiIuLg6jo6PIz89HbW0teDwexGIxOx59cFi+fPmi79na2orOzk68+uqri77WXDQ1NcHNzc0oHZKF\nQiEuXryIu3fvoqurC2vWrDH4PcjiUQ0PMTlra2twOByo1WqjNTIzNZVKhdOnTyMtLc3sipSNQb/d\n/MkGhMDj08arqqrYbsI6nQ79/f1sd+iOjg4MDw/j448/xvT0NEQiEdLS0lBZWYnQ0FCDPEhVKhW0\nWu2i6mvq6upeeDK6QCDA/v378dVXX0EsFrO9kBZC39vl3LlzOH/+PPbs2TMj9GRnZyMiIsLoOxvt\n7OywevVqJCUlobm5GeXl5cjNzUVsbCxWrFgBmUyGixcvYnx8HCKRaFH3qqysnLVztrE4OjoapY6n\ntrYWFy5cgE6nA/C4/o2YJ1pwJEvCkpoPMgyDixcvIiAgANHR0Us9HJOwt7eHjY0NBgcHZ7zu6uoK\noVCIjo4OAI+XTTw8PLBixQrs2LEDqampcHBwwNjYGPz8/GBnZ4ezZ8/i4cOHmJqaglwuR39//6Ie\nGgqFYlHHawwPD2N4eHjGwa7P4ubmhh07duDMmTNsT6OF4nA42LlzJyYnJ3HlyhX2a1BbW4u+vr5F\nNyqcDy6Xi5CQEBw8eBBHjhyBRqPB3/72N5w7dw5eXl7sCepFRUVs5+35GB8fR1tbm9GPq3iSsQqX\nPTw84OjoyC4z6nQ6aLVag9+HLB4taZElUVdXB4lEsuidNObgxo0b6Orqeuq3ckvX1dUFPp8/owEh\n8Hg7d3d3N7slnGEYtLa24vz583jw4AFSU1PB5/MhFAoRFhaGjo4O7NmzB3w+H+3t7bhx4wauXr2K\ntrY2DA0NQavVQigUzrlupaurC8PDwwvuwVNRUQEHBweEhITM6f2urq7QarUoKipCdHT0oo504HK5\nCA0NxdWrVzE1NQVnZ2ecOnUKe/fuXbKfFTs7O8hkMiQkJECj0aCpqQkNDQ3g8/m4fv06enp62I7f\nc1VWVgaRSLSgE+gXyhi9eIDHy1krVqyAVqtFZ2cnGIbBK6+88rM6K/BlQYGHLInm5mY4OTm99Ms/\nbW1tyM3NxaFDhwxeDGnuxsbG0NPT81QwEAqFKCgoQGJiItrb23HhwgXcu3cP69atw+bNmyEWi+Hr\n64tLly6hqakJBw4cgK+vL3x8fBAWFoaVK1di+fLlEIlEGB4eRk1NDXJzc1FbW4ve3l5MTk6Cz+dD\nIBDMOovT2toKhmEW3IMnOzsbq1evnlfAkEqlaG9vR2NjI8LCwhZVm8bn8xESEoKsrCzcu3cPUVFR\nZjFzqO8MHR8fj5s3b0KhUGBsbAw6nQ4PHz6Eu7v7nOqL9DOiKSkpTxVGG5MxevHocblc+Pv7Iygo\nCPX19bh//z56e3shk8ksqk7xZUc1PGRJCIVCTExMLPUwFmVsbAznzp3Drl27LL5IeTZSqRRyufyp\n193c3MDj8XD8+HEolUokJycjOjp6xuwXh8MBl8uFra3tUzNEwOMlz9DQUPaoCq1Wi97eXnR2dqK1\ntRXFxcVQq9UzTon38fGBtbX1onrwPHr0aEHHR+gPB/3yyy9RWlq66KJVe3t7xMbG4tq1ayZdypoL\nKysrhIaGoqWlhX1Nq9Xi1KlT2LhxIz755BPEx8fj97///awPe/3RMobY6TUfxlrSepK3tzcuX76M\nqqoqAMDvfvc7/PGPfzTqPcncUeAhS+Jlr+HRarU4c+YMEhISntlXxtJ5eHhgZGQEKpWKPTqjt7cX\nRUVFUCqVcHZ2xpEjR55a4tHpdDh79iyio6PR3t6O6urqFxYr83g8+Pj4wMfHBytXrgTwOHDqi6GL\niorQ29sLV1dXqFQqBAcHY2hoCM7OzvP6Dbuurg4REREL+q3cysoK+/fvx/Hjx+Hu7r6oRnQjIyMo\nKyvDjh07cPHiRbz55ptmdfZcREQEGhsb2TosOzs7qFQqfPzxxzh//jwyMzOxZcuWWWt05HI54uLi\nTD7zYW9vj8nJyUWdev8iOp0OpaWlbE3ThQsXKPCYEQo8ZEmIRCJ0dXUt9TAWLD8/H7a2tkY9/8fc\n8Xg8eHl5oaurCw4ODiguLkZ7eztWr16NdevW4cyZM7PWNBUUFAAAUlNT0dvbix9++AEhISHz3lVl\nb2+P8PBwtiZDo9Ggt7cXp0+fRn9/P7766itotdoZp8R7e3s/82HHMAxqa2uxe/fueX4l/sHBwQF7\n9+7FyZMn8c477yyo8SHDMMjIyMCqVasQHR0NGxsbnDx5EocPHzbKluqFCAwMBJfLxY4dO2bMpvX3\n9yMjIwMCgWDWgDYxMYGWlhZs3brVlMMF8HgWztHREQqFwuBb+xmGwcDAAGpra5GYmIiqqipwOBy8\n/vrrBr0PWRwKPGRJiESil3ZJq66uDg0NDXj33Xd/9uvzYrEYeXl5GBsbQ1JSEnbs2AFra2swDAM+\nn4/u7u4ZD76amhrU19fj2LFj4HK58Pb2RkhICIqKivDaa68taix8Ph8SiQQajQbp6ekQCoVQKBTs\ndvi8vDw8evSI3UKuD0H6U8R7e3vBMMyil1qkUumijp+4ffs2dDodkpKSAAAhISEYHx/HiRMncOTI\nkUVvBzcE/anvdXV1WL16Nfu6VCrFhx9+iI8++mjWZcXq6mqEhoYu2WG6+mUtQwWea9eu4d69exgY\nGIBWqwXDMPjFL36B5ORkeHl54de//rVB7kMMgwIPWRJCofClXNLq7+9HdnY2Dh48uKg+Ly87hUKB\na9euoba2FkKhEL/+9a9nPMQ4HA57tpY+8PT09ODKlStPFXinpKTg008/RWxs7KIb+E1NTUGj0bDX\nd3R0hKOjI7sbaHp6Gj09Pejo6EB9fT1ycnLA4XAglUqhVCohlUqh1WoX3cl4+fLl6O3txdmzZ3Hg\nwIE5794bGBjAtWvXcPTo0RkfEx8fj7GxMXz//fd4++23zaJ/VWRkJPLz89nAc/v2bVRUVMDNzW3W\nsMMwDORyObZv327qobIMXcfT3t6O3t7eGa85ODhg1apVsLW1/dn/QmRufj57aIlZeRlreNRqNU6f\nPo2UlBSjnuxszsbHx/Hjjz/i888/h0AgwLFjxzA5OTnrAzg8PBz19fVgGAZKpRKnTp3Ca6+99lSR\nskAgQEpKCrKyshbdtE1fsPysB42VlRV8fX2xevVq7Nu3D7/97W9x5MgRhIaGoru7G93d3fjv//5v\nHD9+HFeuXEFdXd2C++ts2rQJwOPTwOdCp9PhwoULWL9+/axLV/pZgzNnzphFnxc/Pz+Mjo5iaGgI\nN27cQH5+/nPf397ezobLpWLowDNbl2g/Pz/weDxoNBqD3YcYBs3wkCWhX9L6aadec8UwDDIzMyGR\nSCz20NPnUSqVuHHjBiorKxETE4MPP/yQXVoRCATo7+9/qsWAp6cnOBwOuru7kZ+fj8jIyGf2XVm+\nfDnkcjmqqqoW9fWd7w4tDocDZ2dnjI2NwcnJCR988AHUajW6u7vR2dmJu3fvIjs7m10u0y+FeXl5\nvbDfDpfLxZ49e3D8+HF4eHi8sDD7+vXrsLW1xYoVK5451i1btuD06dO4dOkSdu7cuaQ/O1wuF2Fh\nYSgoKEB9fT37ularhU6ne2pWa6mKlZ/k5OSExsZGg1yroaEBmZmZiIqKQk1NDYB/bN2fmprC1NSU\nQe5DDIcCD1kSfD4ffD4fKpXqpVgaKisrw8DAAI4cObLUQzEplUqFW7duoaysDOHh4Xj//fef2oIv\nlUrR2dn5VODhcDgIDw9HdnY27Ozsnru9Wv8wP3HiBEJDQxfc02hkZGRBLQJqa2vZHUXW1tbw8/Nj\nj3FgGAbDw8Po6OhAR0cH7ty5g6GhIXh6es4IQbMdgPnT4yckEsms9+/p6cHt27fx3nvvPTcQcLlc\npKen45tvvkF+fj5SU1Pn/bkaUkREBLKysrBixQrI5XL2eAWFQgFnZ2f2fZOTk2hsbERaWtpSDRWA\nYWZ4GIZBcXEx7ty5gzfeeAN2dnZobGyESCTC0NAQxGIx+vr6oFQqDTRqYigUeMiS0S9rmXvg6ejo\nYOsqLO0E9GdRq9UoKyvDzZs3ERQUhKNHjz5zh5BEIkFHR8esMzN8Ph+9vb347W9/+8I6Fk9PT0RE\nRKCgoADbtm1b0LgX0oNHp9Ohvr7+mWGWw+HAxcUFLi4u7MGmU1NT6OrqQmdnJ6qqqpCZmQlra+sZ\nxdAeHh7g8Xjs8ROnT5/G0aNHn2q2p9FokJGRgU2bNs2pEZ+VlRXeeOMNfPHFF3BwcGC36S8FX19f\nqFQqREdHo7q6Gnw+H3FxcU8VJd+9e9coJ5XP12IDz+TkJC5cuAC1Wo1jx45BIBDgyy+/xPr167Fy\n5Uo0NjbC19cXg4ODtKRlhijwkCWjDzwL2bprKhMTEzh79ix27Ngx4zdWS6XRaFBRUYHS0lIsW7Zs\nTlurpVIpysrKnnq9u7sb5eXlsLOzw9jY2JweduvXr8enn36Krq6uBfWdUSgUz5xFeZa2tjY4OjrO\na8u3jY0NAgICEBAQAODxb/2Dg4NsX6DKykqMjIzA29ubbY64fPlynDp1Cu+8886M4FxcXAwXF5d5\ndVO2s7PDwYMH8cUXX5j8iIYncblchIeHo6SkBF5eXrCysnpqFkdfrLx58+YlGeOTRCIRVCrVgnrx\n9PX14dSpUwgODkZqaip4PB4KCgogEAiwcuVKcDgctlEmn8+nwGOGKPCQJWPuW9P1DfJiYmIW1UTu\nZaDValFVVYWSkhJ4enriwIEDs3ZAno2DgwNKSkrYLd+7d++Go6MjTp8+ja1bt7I7ojw8PF54LVtb\nW2zcuBFZWVlP7VSai4XM8NTW1i46MHA4HIjFYojFYsTGxgJ4vBzY1dWFjo4OVFRUsOcs/eUvf0FS\nUhJ8fX0xNTWF6upqvP/++/OubXFycsKbb76Jb7/9FnZ2dnM67NQYIiMj8e2330Imk8263buzsxMa\njQbLli1bgtHNtNBePHV1dcjOzkZaWhobTB88eIA7d+7MugxJgcc8UeAhS8bct6YXFRWBw+HMuhPD\nUuh0Oty9exdXr16Fi4sL9u7dO68ZkhMnTuCDDz7A9PQ01Go1rK2t8bvf/Q5xcXH44x//iLCwMNjb\n2+PixYtYv379nK4ZHR0NuVyOyspKJCQkzOvzmW/g0Wq1aGhowHvvvTev+8yFra0tAgMD2U7cDMOg\np6cHp0+fxp07d1BWVobBwUG4u7vj9u3b7EzQfJZ9PD09sWfPHpw9exZvvfXWnEKloTk6OmJ6ehrj\n4+OzzlKZQ7Hyk5ycnDA8PDynwKPT6ZCfn4979+7h4MGD7O5MpVKJjIwM7NixY9a+SHw+3yx20pGZ\nKPCQJWPOW9MbGxtx9+5dvPvuuxZ5AjrDMKirq0NxcTGEQiF27NjBFunO1fnz5/Huu+9icnKSfU2l\nUgEAKisr8de//hVpaWnw8fGBWq1Gf3//nJYvORwOXnvtNXzzzTcIDw+HUCic03h+2oNnLlpaWuDm\n5maSs9A4HA68vb1x5MgRHD9+HJ6enuzurY6ODty6dQtdXV2wt7dna4EkEgnc3Nye+z3o7++PzZs3\n47vvvsORI0dMfqp6fX09xGIxent7n2raqFKpcO/ePbNqwDfXOh6lUomzZ8+Cw+Hg2LFj7PcVwzC4\ndOkSwsPDIZPJZv1Y2pZunijwkCWj39VgboaGhnDp0iW88cYbc37YviwYhkFTUxOKiorA4/GwefNm\nBAQEzPu3b4Zh8Jvf/GZG2HmSWq3GlStXUFNTg6ioKLYJYXJy8pyu7+HhgejoaOTn52PHjh1z+hiF\nQsF2TZ6rurq6Wc97MiZ9Y7rc3Fz88pe/hI+PD3uyu06nw6NHj9ju0Ddu3MDExAR8fHxm7Aj7aVFw\nZGQkxsfH8d133+Hw4cMmLQ6uq6tj63h+OttRU1ODwMBAs/o5mkvg6enpwalTpxAZGYkNGzbMCJyV\nlZVQKBTYs2fPMz+elrTMEwUesmTMcUlrenoap0+fRnJy8ryLX80ZwzC4f/8+CgsLodFosH79eoSE\nhCx4maGsrAwKheK57xBr91IAACAASURBVFGr1Th+/Dg++eQThIeHIysra86BB3jc1O3TTz9FR0fH\nnJrVzXc5a3p6Gk1NTSbfKq1SqXDz5k2sXLkS58+fn3H8BJfLhaenJzw9Pdl+PBMTE+js7ERnZydK\nS0vR3d0NR0fHGSfFi8ViJCYmYmxsDD/88APeeustk+woHB4exuDgIFauXAk+n4++vj522UdfrLxx\n40ajj2M+XtSLp7q6Grm5udiyZQt7Tpveo0ePUFRUhMOHDz+3GzcFHvNEgYcsGXNb0mIYBllZWXBz\nc5t37Yg5e/jwIQoLCzExMYFXX3113qeB5+Xloba2FjKZDEFBQfDz80NfX98Ll/q0Wi3a29sBPD5z\na2xsDJcuXYJOp8POnTtfeF8bGxts2rQJWVlZc1panG/gaWpqgo+Pj8lnH65cuYLg4GCkpaWBYRic\nO3cOb7755jM/P6FQiJCQEISEhAB4PAvU19eHjo4OtLW1oaSkBCqVig1AVlZWOH36NN544w2jL8fW\n1dUhLCwMfX198PT0RG1tLRt4enp6oFKp2J1s5uJZMzxarRY5OTlobW3F22+//VRfKY1Gg3PnziEl\nJeWF9T8UeMwTBR6yZMwt8MjlcnR3d+Po0aNmU2C5GF1dXSgqKsLg4CCSk5MRHR29oAegUCjE2NgY\n5HI5qqur2WJMfZO5Z+Hz+bC3t8d//dd/QaPRgMPhoKqqal7b+yMiIlBZWYny8vIX9puZb+BZiuWs\nhoYGtLe34/333wfw+PiJ7777Dvn5+exRFC/C5XLh5eUFLy8vvPLKKwAeH/mhXwbTaDRoa2vD//zP\n/yAiIoKdBXJxcTH493VdXR02bdqE0tJSRERE4NatW9i4cSM4HA4qKysRGxtrdj9LswWe8fFxnDlz\nBra2tjh27Nish5vm5eXN2IX3PBR4zBMFHrJkhEIhlEqlWRwv0d3djcLCQhw+fBjW1tZLOpbF6u3t\nRXFxMbq7u7Fu3TrExsa+8BiEJ6nVavT09KCrqwtdXV14+PAhe8YVwzCwt7fHm2++iS+++OK5gdXK\nygoffvghbt26BYVCwQYknU6HGzduYNmyZfD09Hzu2PQFzF999RUiIiKee1L4yMjInHv3qFQq3L9/\nf8ENDhdiYmICWVlZ2Lt3L/s99tPjJ/SNDedLJBIhNDSU7QOjVCrxxRdfYGRkBCqVCsXFxVCr1TOK\noX18fBb1vT4wMIDx8XH4+vrizJkz2L59O8rLy9HV1QV3d3fU19fjgw8+WPD1jUUkEmFqaortxdPR\n0YGzZ88iNjYWycnJs/5b1NTUhMbGxhd2wtajwGOeKPCQJcPj8WBjYwOlUrmkRY2Tk5M4c+YMtmzZ\nMq/eHOZmYGAAxcXFaGtrw5o1a5Cenv7COg59kaw+3HR3d2NoaAju7u7w8fFBSEgI1q5di7///e//\nz96bBzWd7+n+ryQECPsOIvu+uYCi0qK22rZbiytH26VdGvu05/aZO3VqamrqTN25Z6rmzq26M2fp\n04s9R1u7td1FAVEUFVxQdhAFBERWAdllCyQhye8PK98yshiUbr33l6cqVQrJd0vI5/m+38/7eZBI\nJAQGBrJu3TqkUilff/018fHxo2YGyWQyli1bRmZmJq6urkilUjo7O9FoNPj4+NDd3U1JSYlAUry8\nvPD29hZaMi/C2dmZmTNncvXqVdavXz/muUykwlNZWYmPj88v5vKta5dOnz4dLy8vvd+9GD/h6Og4\nKdoxCwsLdu3axaFDh5g/fz6bNm2itbWVpqYm2tvbycjIoLW1Vdifp6cnnp6e4wavvgydWLm3txcT\nExNsbGwIDw+nrKyMtrY2vL29R43beNvQefF0d3fT0NBAZmYmcXFxQsvwZehasfHx8QZ/XoyE592E\nkfAY8Vaha2u9LcKj1Wo5d+4cISEhIwSK/7egu7ubmzdv8ujRI2JiYoiLixv1zl2r1dLT0yOQm6am\nJp4+fYqNjQ1Tp07F3d2dWbNm4erqOkKQ6efnh7+/v+AoC89jBbZs2UJSUhIajYahoSFEIhEikYht\n27bx1Vdf8eWXXwo6Ht3rZDKZIBQeHByksbFR0Bm1trbi4uIiECAvLy9kMhmLFi3im2++ob6+fkwD\nu4nkaJWWlk7I2fhN8eDBAzo6OtiwYcOov3d2diYuLm7M+InXgZWVFdu2beOHH35ArVZz8+ZNvLy8\n2Lx5M/Bck9LS0sKTJ0+orKzk2rVraDQaPTG0u7v7qKRZq9VSWlpKXFwczc3Nwjh6eHg4P/30E9bW\n1hMSqP/SsLW15cqVK/T19bFnzx4cHR1HfZ5WqyUpKYlZs2ZNyDjROJb+bsJIeIx4q3jbbsu3bt1C\nqVS+c5MkhqC3t5dbt25RXl5OdHQ0v/3tb/W0B52dnfzHf/wHvb29zJ07l/b2dkQiEVOnTmXq1Kks\nWrQId3f3UfUKL2Pbtm16/29oaCA9PZ3/83/+DwcOHCA1NZXS0lIePHjA119/LQg+165dS2JiovDl\nHxwcLBAgeE5+goKCBCdrlUoltNHy8vI4d+4cdnZ2eHl5ERoayoULF9i3b9+INpiuRWEIcZbL5TQ2\nNhIfH//K574J6uvrsbGxQSKRcOXKFbZv3z7uZE9wcDBtbW2jxk+8LhwdHVm8eDEXLlwAnrsD61rI\nJiYmQmUHni/uvb29Qkhqenq64J304ki8ra0tbW1tqFQqPDw8uHbtmkB4XFxcMDEx4dmzZ2N61Lxt\n9PT0CET/008/xczMbMzn3r17F5VKNWHyZjQefDdhJDxGvFW8zdH06upqCgsL2bt374Q0Lm8b/f39\nZGVlUVJSQlRUFF988QWmpqY8ffpUr3pz4MABKioqgOcE5fjx40KLobe3l87OztdaVJ89e8aZM2dY\nv369QGzWr1/PunXr+Pd//3c9UXJwcDDOzs60tLQgk8loaGhArVaP2X6SSqV6SeUajYanT59SX19P\nfX093d3d/Od//idBQUFCFcjR0ZGenh6D2zHl5eUEBAT87FqtCxcu8OzZM+zt7YmOjhaml8ZDbGws\nbW1tpKamsm7dujfWtjU1NXHx4kXh/7oJr9FiQ3StHltbW0HMrVKpaGlpobGxkbKyMi5fvoxIJMLM\nzAw7OzthXH7BggXCdmQyGTKZ7J007KyrqyMxMZEpU6bg4uIyLtlpbm7m7t277N27d8LnYmxpvZsw\nEh4j3ire1qTWs2fPSEpKIj4+/p3UGYyGwcFB7ty5Q1FREQEBASxatIjOzk6OHTtGe3s7jo6OTJ06\nFV9fX2JjY9m/fz8qlQp4/kV/+vRp5HI5fX19wPOWxn//7/99QpNNCoWC48ePExsbO+IOXiQSYW9v\nT1dXlxBxIBKJWL16NUePHuXTTz9lYGCAn376iTNnzrBjx45XVpfEYjHu7u64u7sTExNDR0cH33//\nPc7OzjQ0NHD79m1UKpUQ/Nnc3Iybm9u4C1RZWdkvkjDe29uLWq2mo6ODmpoaZs2a9crPmkgkIi4u\njsOHD3P37l3mz5//Rsdga2vLtGnTKCsrQ6vVolKpKCsrMzgnTSqV4uXlJeiOtFot3d3dfP/997i6\nupKWlkZLSwsZGRk8fvwYNzc3Ojs7kUgkaDSad4b0aLVacnNzycrKYv369QwODvLw4cMxn69UKklM\nTGTVqlWv5VwtFovRarXv1DUwAiR/+MMf/vC2D8KI//+itbWVgYEBIW/ol8Dw8DDHjh1j1qxZTJs2\n7Rfb7+uio6ODS5cukZqaSldXFyqVSohwcHV1JSoqiuXLlzNnzhyCgoKYMmUKlpaWeHh4cPbsWaRS\nKZ9//jm9vb0MDQ2h0WjQaDRIpVIWLlw4bpvlRWg0Gs6cOYOLiwuLFy8etfrw+PFjbGxs9CIkbGxs\niImJwdLSEltbW5ycnHjw4AFFRUW4ublNaEGxsLBAqVQKeph58+YRERFBa2sr/f39VFRUcP36derq\n6oTRY0tLS6GC19fXR2ZmJh999NHPuhDpyKluuq23txdnZ2eDqjwSiYSAgABSUlJwdnYeU19iCExN\nTQkJCWHevHnY2toKuq2cnBzu37/PjBkzJlTdFIlEdHV18ejRI3bt2oWvry+VlZWsXr2awcFBioqK\n6OvrE0bjdZ9TS0vLMa/34OAgAwMDBrVWXwcqlYrk5GRqamrYsWMH7u7uKJVKSktLmTVr1qivSU1N\nxdbWVq9yNRGIRCKysrKIiYn5v6p6/P86jBUeI94qLC0taWtr+0X3eeXKFWxtbYmJiflF92sIFAoF\nzc3NNDU18eTJE2pra1EqlVhbWxMVFUVAQABTp041KDogJiaG//2//zdSqRQPDw/hLl8HtVrNn/70\nJyHaICgoaNzFNT09neHhYVauXDlmq8XBwWHUuJAXv/SDgoJISUnhgw8+IDExkWnTprFkyRKDideC\nBQv49ttvqampwc/PD1tbWywsLJg+fTqxsbGCRqe+vp5r167R1taGq6srXl5eDA0NERAQYPC+JgK1\nWi2cp24iTSKR4OrqykcffWQQ2dHB1taW+Ph4Tp06xe7du994etDU1JSoqCiioqKYPXs2paWliMVi\nsrOz+eGHHya0rdLSUiIiIhCJRDQ3NzN16lShDfno0SM2bdpEY2MjLS0tdHR0cO/ePbq6unBzc9Ob\nCNNZDNy5c4ecnBw2btxIaGjoG53ny+ju7ub06dM4OzuzZ88eoYU7XrxEaWkpjY2NfPbZZ2+0b6OO\n592DkfAY8VbxS7e0SkpKqKmpYe/evW/d+0etVuuNhDc1NfHs2TNcXV2RSCS0trbi5eXFsmXLRri+\nGoLq6mpCQkIoLS2lrKwMHx8fwZgOYPHixcyZM4eamhqqqqq4e/cuZmZmAvnx8vISFvDCwkKqq6v5\n9NNPx71jdXR05MmTJ+Mel0QiITg4GIVCweeff87Fixc5cOAA69evN6jVIpVKWb58OWlpaXz++edI\nJBJ6enoE4ayFhYWeM7FSqRSE0IWFhajVavbv3683Cfa6U1GPHj3iT3/6E0ePHmVgYABTU1M++ugj\nYXR/w4YNrx3h4eXlxQcffMDJkydJSEiYlAqIVquluLhY8ETKzMyc8OvLysrYsWMH8LyFqCNybW1t\ndHd3ExQUhJubGwcPHmTHjh2IxWIUCgVNTU00NjZSVFRESkoK5ubmeHh4CLquc+fOERMTw+LFiwHI\nzc3F2tqa8PDw1zrXx48fc/78eWJjY/WmC+H5jZZSqUSpVOppuZ49e0ZaWhrbtm17Y42XUcfz7sFI\neIx4q/glCU9rayvp6ens3LnzZyufA3z55Zf867/+K2ZmZhw8eJDVq1cL2ocXyU1rayt2dnbC1NTs\n2bNpbm4mKysLNzc3du7cabDWYjQ8evSIuXPnUlRUhFarpaGhAbFYjFgsxt7enuzsbCwsLIiMjCQk\nJAStVsvTp0+pqqri2rVrdHV14efnh729PcXFxezZs+eVPiQODg7cv3//lccWFhbGnTt3mDt3LvHx\n8dy/f5+jR48yb9485s+f/8p2U3BwMEVFRWRnZxMbGzuuB4+pqSm+vr7Y2dmRn5/P7373O9rb26mv\nr6e0tJRLly5hZmYmkB8vLy8cHR1fSVISExP55JNPUCqVwsKmVCpJSkri8uXL/P73vxeMAF8XkZGR\nPH36lMTExDeOitDlqX344YdkZWUxPDxMaGgobW1tBhPqhoYGZDKZ0LJsbm4WJpiKioqYOXMmEokE\ne3t77OzsqK2txd/fHzMzM/z8/ISYCa1WS2dnJ/X19ZSVlQHPW823b9+mvLycyspKDh8+jEaj4dtv\nv2XXrl0TOs87d+6Qm5vLpk2bBAH8i9AJtHt6erC3t8fExASNRkNiYiLz588fkfr+OjAxMRE0dEa8\nGzASHiPeKn4pwjM0NMTp06dZvnz5a1VLDEV+fj6///3vkcvlAGzcuJG//OUv9Pb2IpVKBXKzdOlS\npkyZgpmZGRqNhgcPHpCYmIi9vT3x8fFvbD6na405OTkJniBqtRq1Wo2DgwP79u2jo6OD5ORkysrK\nWLNmDXZ2dkJkwaJFi+jv76e4uJibN28iFotJSkoSqj+urq6jEoKxWlovw8/Pj/Pnz9PX14e1tTUz\nZszA29ub5ORkHj16xLp16wQh8mgQiUSsWLGCgwcPMm3aNINMB3W5T6ampsL78N5776HVauno6KC+\nvp66ujpu3brF8PCwQH68vLxGCKGLiorYsWPHqGnxGo0GuVzOv//7v+Pv78+WLVteeT3Gw/Lly/np\np58mFD+hg1KppKamhrKyMqqqqlAqlSQkJLB8+XLc3NwIDQ3lxx9/ZM2aNQaRs9LSUqHiopugmzJl\nCsPDwzx48ICEhAThueHh4ZSWlo6qzxOJRDg5OQlE0czMDJVKhbW1NY6Ojpw4cUKwqzh8+LDBhEeh\nUJCcnExvby979+4dtXL38OFDIfH8b3/7G8PDw3z22WdUVFRgZmY2aa1uY0vr3YOR8BjxVmFhYSEI\naX8uEalWqyU5ORk/P7+f3Wyuurpar+Wjyz3atm3biAkdXXvgxo0byGQy4uLiRr0bfR3U1tYydepU\n5HI5arUasViMqakpH374IREREYK2JCEhgbt373LgwAHef/99Zs+eLRAZiUTCvXv3WLVqFdOnT6e+\nvp6qqipOnz6NWq0WyI+vr6+gjbCxsWFoaGhEq+BlmJiYEBgYyMOHD4U8KDs7Oz755BNycnI4ePAg\nS5cuJSoqaly90Jw5c0hLS0OpVL7Sg6e0tJQVK1aM+LlIJMLZ2RlnZ2chofzZs2c0NDQIbbC+vj48\nPDyENtgf/vAHQZA7FuRyOb///e/ZvHnzG7VP3yR+4vLly5SUlAgtLLFYzOLFi/n2229pbW1ly5Yt\nODk5cfr0adra2liwYMGYx6rRaHj48CF79uwBnuuUrKyskMlkPHjwADc3Nz1LgvDwcP7rv/5LT9v0\nMiwtLVm4cCFeXl54eHgIn5n09HROnjyJRqMx+Hw7Ozs5deoUHh4ebNiwYUyd1uDgIDU1NYKeTSqV\nCoJrQ6MjDIGxpfXuwUh4jHirEIvFyGQyBgYGfrbx8Lt379LX18fGjRt/lu2/iAULFgjGbhKJBEdH\nRx48eMD9+/cRiUTCqOqcOXOor69HIpGwfPly/P39J1VT9OjRIwICAnBycmLNmjV4enpy8OBBQkND\n9bx3xGIxsbGxBAcHk5KSQnl5OWvWrMHW1pbTp08TFBREVFQUAP7+/vj7+7NixQo6OzsF3U9iYiLe\n3t4EBQURGBgoVHle1Y4LCwsjNzdXIDzwnHzExMTg7+/P+fPnqaqqYs2aNWNmaM2fP5+vv/4aCwuL\nca9fe3s7crl8RKzDWLCzs8POzk4gyHK5XCBAycnJXLp0SU8APhba2tooKioacxrIUFhYWLBlyxZ+\n/PFHnJycDM4M++CDD6iqqhKqJe7u7lRVVaHRaOjr66OhoQEvLy8SEhI4deoUbW1tYzp119XVYWtr\nq2cBoGv9FBUVCWRRB91E3uPHjwVjyZdhbW3N+++/P+Ln33//PRYWFoSEhCAWi8nKyiI2NnbM86ys\nrCQlJYUlS5a88lpHRkZSXFxMU1MTWq2WoKAgkpOTx/2cvQ6MhOfdg3Es3Yi3jvv37+Pv7z+pXzY6\n1NXVkZ6ezo4dOwyabHpT2NjYEBcXx+DgIOHh4fzP//k/8fX1pa6uDo1GIyySQ0NDLFmyhA8//NAg\nvchEoNVquXTpEosXLxbaVBYWFoKYeDQiYmlpycyZMxkaGiIpKYmqqirMzMxYu3btiGMTiURYWFjg\n6enJzJkzmT17NqamptTW1pKens7Q0JBgBGhlZTXmudna2pKens7MmTNHLLCWlpZERkbS3t5Oamoq\n9vb2eqPuOkgkEoaGhqitrSUmJmbMKmFeXh4ODg4EBgYadA1fhlQqxcnJiYCAAOzt7Tl06BBKpfKV\nr5PJZCxevHhSpo8sLS1xcnLi/PnzREREjGuap0NtbS3l5eWCRuX999/n+vXrDA8PC628qKgozMzM\nmD59OtXV1WRlZREYGDhC53b79m18fHwEZ+bi4mKcnJywsLDgzp07rFmzZsT1VygU1NTUTPj8BwYG\nuH//PpaWluzYsYO0tDTEYvEIoqfVarlx4wa3b99m8+bNBrXlRCIR/v7+5Ofno9VqsbCwwMfHh3nz\n5k3oGF+F+/fv4+XlpVf1MuLtwuiIZMRbx8/lttzX10diYiLr1683OGNpMhAREcGhQ4dYt24dd+7c\nwcrKSq9VJRKJ8Pb2HlMH86bQRUi8PMocEREhCERHg1gsJiYmhqioKFpbW5HL5QbpcWQyGREREaxf\nv55/+Id/IDAwELlczoULF/jjH/9IUlISZWVlI1pAUqmUwMBAwQ36ZUgkEpYuXcrmzZu5evUqSUlJ\no7aRLC0tsbCw4O7du6NuR9c6nCzPJZlMNiFtxmQGlAYHBxMdHc2pU6deWT148OABKSkpbNu2jb17\n9+Lt7c3g4KBe2OuTJ09oamoCnlck1q5dy/Tp0zl48KBeBIharaaiokJvYkpX4SkqKhrTzyc8PJyq\nqqoJVzquX78OIHz+duzYIbiL6zA0NMSJEyeoq6tj7969E9K96WwpRCIRQ0NDLFu2bELHZwiMGp53\nD8YKjxFvHTU1NVhaWr7RRNLLUKvVHD9+nGnTphEZGTlp250IampqaG5upqqqCrlcLggz9+zZQ19f\nH6mpqbS1teHo6Dip4aklJSXIZLIR6c92dnZcuXKFyMjIMfU1VVVV3Lp1i88++wypVMr58+cB8PDw\nMIic6RYQlUrF9u3bCQ8PR6lUUl5ezpUrV6itrUUul2NhYSFU3HQGeGPB1taWyMhIampquHr16giz\nwrKyMlxcXCgoKCA8PHxEZeLp06eUlpbywQcfTArBtLGx4cCBA/T09LzyuWKxmC+//NKgaoyh8PLy\nora2lkePHhESEjLqORUUFJCZmcn27dtxd3dHJpMxY8YMTExMsLe3Z2hoiKlTpzJjxgy8vLyE4xOJ\nRHh6euLi4kJiYiJmZma4u7tTXV1NZ2enUAVRq9VcuXKFpUuXkpqayqpVq0atoJqZmVFdXY2FhYXB\nXkIdHR2kpaUJuiONRsO0adMICAjg/Pnz2Nvbo9VqOXLkCO7u7mzcuHHCU5eDg4McOXKEgwcP0t7e\njq+v76Tp53QoLy/HxcXljT2UjJg8GAmPEW8dT548QavVGqyvMARXr15FpVKxevXqt+K309raSkZG\nhl7bw8vLi7lz5xIWFoavry+zZs3i2bNnXLp0icbGRuzt7SclJTsjI4MZM2aM+KLVefsolcpRNSCt\nra2cPn2azZs34+rqytSpUwkLCyMnJ4f8/Hw8PT0NImYKhYLy8nIiIyMxNzdn6tSpTJ8+nblz52Jp\naUljYyMZGRkUFBQgFoupqqpi5syZ45ICnXePg4MDSUlJ9Pf34+3tjVgspqioCF9fXxwdHSkuLiYi\nIkJoHYpEInJycnB1dRVGot8UIpEIqVTKzZs3xx07lkqlbNmyhV/96leT+hkUiUQEBgaSlZWFUqkU\nWkw6ZGVlkZuby86dO0e0Aa2trfHy8qKpqUlo44x23R0cHAgODiYtLY2Ojg6ampoE00t4rk2qqanB\nzs6Onp6ecSebVCoV1dXVhIWFGXR+qampdHR0CO9ha2sr8+bNw8bGBl9fX06dOkVxcTEffPABCxYs\nGHfY4eHDh9TV1eHs7CxUoLRaLQsXLuTs2bP09PRQW1vL6dOnCQ8Pf2MbgRdRUVGBg4PDzzoVasTE\nYCQ8Rrx1tLa20tfXN2npymVlZeTl5bF9+/ZJSZyeCHQxELdu3cLMzEwYWzY3N2fWrFl6Al0TExO8\nvb2Jjo5mcHCQy5cvU11dLQQ4vs4iqVAoSE9PZ9WqVaO2GCQSieCX8iIGBgY4cuQIy5Yt09O5yGQy\nQbiblJSEWq3Gw8Nj3EVGJzJ97733RuzbycmJ4OBg5s2bh5eXF319fTQ3N5OdnU1ra6swmjxWBcrR\n0ZEZM2bw4MEDsrKy8PT05P79+0ybNo3Q0FBu3brF0NAQ58+fF0bLL1y4wNKlSydVI+bn50dGRgad\nnZ2jtmukUinu7u6sWrWK69evU1xcTG1tLe3t7ZiZmb2xQF8ikRAYGEhKSgouLi44ODig1Wq5fv06\n5eXl7Ny5c9wx/crKSmxtbcd1f9a5VxcUFFBVVcWyZcsEwqsTPtfX1xMZGSlkp40GW1tbLl++zNy5\ncw2KWbCysmLKlClUV1cTERFBcHAwXl5eiEQicnNzaW9vR6vVEh0d/cpW9e3bt8nOziYnJ4f+/n6c\nnZ3Jy8vjP//zP/VuRlQqFQUFBfzd3/3dK4/PUFRVVWFjYzOplWsj3gzGKS0j3jqsrKxoaWmZlG3p\nCMf27dsnVTvxKnR3d3Pr1i2qqqqYN28ecXFxJCcnY2lpiYmJCT4+Pnpk50VIpVLmzp3L7NmzuX//\nPqmpqchkMhYsWEBgYOCEiI9uHH0swuDv7y/4lOiqScPDw5w8eZLp06ePqnMRiUTMmjWLgIAAUlNT\nOXjwIGvXrh3zi9za2hqlUolCoRizaiMSiQTPH0dHRwoLCwkMDKSqqorLly/j7Ow8puePpaWlnlmh\nSqXCxsYGuVyOTCYT3IM7Ozt58uQJpqamk3KXPTg4SEVFBQUFBTQ3N5OQkEBeXh4//vgj8LzNI5VK\n0Wq1LF26lB9++IH+/n6OHDlCb28vvb29VFdXC4Gob4oX4yd27txJXl4eLS0t7Nq165UCfUMmzOA5\nUY+KiqKzs5MTJ06wZcsWXFxcaG5uxtbWlurq6lcKkq2srHB3d+fRo0cGVXm8vb3x9vYmLS2NWbNm\n4ePjg1wu5+TJk2i1Wvbt20dDQwMnTpzgo48+wsbGBoVCofcYGhpCoVDQ0tIiZMfl5eUJ12g0bc2L\nmqXJgM7/yoh3B0bCY8Rbx2SZDyqVSk6dOiWY+v0S6O3t5datW5SXlxMdHc1vf/tbQU8QHx8PQEtL\nCydPniQ2NnbcXmsW1QAAIABJREFUyohEIiEyMpIZM2bw8OFDMjMzycjIIDY2lrCwMIN8inTj6GPB\nxMSE4OBgysrKiImJQavVkpKSgo2NzajjwS/C1taWrVu3UlJSwtGjR5k9ezYLFiwY4XciEomE0XRD\n3ofAwEAuXLjAr371K2bOnIlarX6l549IJGLGjBm4u7uzf/9+zpw5g0KhoKOjQ9huX1+fYJT3pi2l\ngoIC0tLSkEgkqFQqxGIxS5cu5dNPP2XNmjVkZmbi6upKb28v8+fPZ+XKlQA4OTnh4+NDbW0t8FzT\n8nLl603g5eXF4sWL+f7773Fzc+OTTz6ZVL0QPNei6EJmdSaFLS0tODk5MW3aNINyyV40IXyZnLxI\nUF586CpWurBY3X7++Mc/IpVKkUgkJCYm4uLigpWVFWZmZpiammJmZoa5uTm2trY4OjrS0dGBSCRC\nLBYTHh5Od3f3qC7IrxthMRaMY+nvHoyEx4i3jskgPFqtlgsXLuDh4SH4xvycGBgYICsri3v37hEV\nFcUXX3wx5l31lClTsLW1pbKy0qDxXN0Xc1hYGNXV1dy+fZvMzEzmz5/P9OnTx1xgtFot1dXVrxyv\njYiIICMjg5iYGG7fvk1nZye7du0yWJQ8c+ZM/P39uXjxIn/7299Yu3btCE2Qg4MDnZ2dBhEeXexA\nZWWlEE2giyFYvny54PmTnZ09wvNHt6+wsDBu3bqFs7MzXV1dDA8P09vbS1tb24RiCcaCn5+fXotS\nKpUKmrP29nb27dtHQEAA3333Hfn5+cyaNUuoKq1YsYIDBw4Azz8Lhw8fZuPGjYLGqqOjg6ysLLy8\nvIiMjJwQOVOpVFRVVWFhYYGJicmEWriG7EehUFBdXc2HH36IWCxm1apVXLhwQZjgi4mJIScnR4+4\n6Kp7o5GYx48fY2ZmNu7D2toaJycniouLcXZ2pry8nCVLlgij+KampgL5LykpISMjg82bN486/i0W\ni4W/uw8//JALFy4QFxdHa2srFy5cYGBgAKlUilgsZuXKlQwODk5aZdhIeN49GAmPEW8dVlZWgjHa\n6yIvL4+Ojg7BBfbnwuDgIHfv3qWwsJBp06bxm9/8xiA9RnR0NPn5+RPyI9GJUwMDA6mvr+f27dvc\nvHlTGB1/uW011jj6y/D19aWnp4fc3FwKCwtJSEiYsNbJ2tqazZs3U1payokTJ5g5cybvv/++QMYM\njZjQITQ0lLKyshHaIt35ODk58d577zE0NER1dTWPHj0iMzNTuAYeHh7s3LmT5ORknJ2daW9vp7u7\nGxcXl3ET4A2FlZUV9vb2KJVK1Go1oaGhQijm06dP8fb2FvLStFotR48eZd++fVhYWODi4kJ4eDiO\njo7ExsZSWFjIoUOHWLJkCaWlpezbtw9TU1PUajXR0dGkpaUZNHWkG8u2tbVl3759nDx5kuvXrwsj\n1lqtFpVKNWpFRac96u3tHfX3uodcLkelUvHNN98IhEQmkwk/7+jowMLCAjMzM6ysrHB0dNQjL+bm\n5sK/z549y4wZMwyyB1Cr1aSmplJbW8uePXvGbEnOmDEDpVLJkSNH2L179wjR/7Rp0/Dx8cHV1ZXs\n7GyGhoZYtGgRixYt4l/+5V+oqqrCz8+Pzz77jJqaGg4dOsT27dsnxcbCOJb+7sFIeIx465DJZCgU\ninEt6MdDY2Mjt27deq2F21AoFApycnLIzc0lNDSUX//61xP6UgwLCyM9PZ329vZRDfReBZ2uQRcu\nmpWVxZw5c5gzZ46wOFZXVxMQEPDKO3exWIy3tzfXr19n9+7dry2gFYlETJs2DV9fX9LS0vjuu+9Y\nu3Ytnp6eODo6Ul9fb/C2goKCuHjxIkNDQ+Mu9ubm5kRERBAREYFGo+HatWvU1taSmppKf38//v7+\nKJVKuru7yc3N5fbt2xw/fpzTp0+/to5HpVJx8uRJnJ2d2bhxIz/99JNgdVBfX4+7uztSqZSuri5B\nGyOXyzl+/Di7d+9GIpFgY2NDRkYGFRUVLFiwgJ07d3Ls2DH+6Z/+CZVKJfgL5ebmsn//fr744otR\nKyS6R19fH/fu3cPCwgJra2vOnDmDUqkkNzeX4uJitFotCoUCExOTUaso/f39woi6paUlDg4OeuRE\n90hNTSUiIkKPiBYUFHDr1i3s7Oxob29ny5YtBv0t6HygXkV4+vv7OX36NAB79+59pR4pOjoahULB\n0aNH2bVrl94koc7+oKWlhaysLBISEhCLxZw4cQJTU1NiYmL43e9+ByCYnx46dIitW7eOK8Q2BCYm\nJgaZUxrxy8FIeIx46xCJRFhaWjIwMDDhseyBgQHOnj1LXFzcz+JoqlQqyc/P5+7duwQEBJCQkDBu\nqOVYkEgkREVFkZeXx+rVq1/7eNzd3fnVr35Fe3s7d+7c4a9//StRUVHMmzePR48eGeQW29vbS21t\nLTKZbFImSKysrIiPj6e8vFwY7w0MDKSzs9PgbZibm+Pj40NVVZXBeWdisRiRSER4eDixsbH09PRQ\nVVXFo0ePePLkCampqahUKmGhO378OBYWFhPKbNORHSsrK+Li4hCLxXqTPI8fPxbG3evr64U7erFY\nTEtLC+Xl5VhZWdHT04NIJKK5uZmzZ88Ko+1SqVRPSzI4OMhf//pXwbdptGoJPJ8AcnV1JSwsTI+o\nDAwMcOHCBSElfKwbiKSkJHx8fEZU1F7E4OAgT548EbRoOtTV1TE0NMT27dspLCzk4MGDxMfHv9JW\nIiQkhLS0tHFJ7ZMnTzhz5gyhoaE0NjYa7I4eGxuLQqHgp59+YufOnXrbVyqVJCYmsmLFCmxtbUlM\nTOTRo0fAc3L14o1WTEwM1tbWHD16dMykdUNhFC2/ezASHiPeCeh0PBMhPBqNhsTERKZPnz7CZO9N\nMTw8TGFhoaCt2LVr12tVZl7E7Nmz+fbbb1m6dOmEjdJehrOzM+vWrePZs2fcuXOHr7/+GpVKxapV\nq8Z9nVKp5OTJk8TExFBYWEhbW9sb38nqEBYWho+PD5cvX+bChQuvDNd8GaGhoTx8+HBCAa/Pnj0T\ndEK2trZER0czc+ZM6urqhEqXWq2mtLSUL7/8ErVajampqUA2fHx8sLa2xtraGisrK6ytrbG0tBQC\nJVNSUpBIJISEhHD//n2Ghob0Hg8fPsTR0ZGysjJ6e3uFCo9Go8HKyoo7d+5gbm6udy20Wi2mpqaj\npojrxND//M//POr5dnZ2cvToUWJjY8cUP4vFYlJSUkhISHij8feHDx8K2qUXUVdXR0BAgFAhcXZ2\n5vTp0yxevHjcHCsdqa2srBzVaLKwsJCMjAzWrFmDg4MDubm5EzreJUuWoFAoOH78ONu3bxfanZcv\nX8bDw4OIiAjOnDlDVVWV8BqxWMyTJ0/w9vYWfhYREYGFhQVnzpxh9erVBvsHvQyjhufdg5HwGPFO\n4HXiJXTjx4sXL37t/Q4NDaHVagWholqt5t69e9y6dQs3Nze2bt06aRNf1tbW+Pv7U1JSwty5cydl\nm3Z2dqxevRp3d3du3rzJ4cOHCQ4OZv78+SO0PFqtlqSkJFxcXIiNjWVoaIjS0tJJIzzwvIWwYcMG\nKioqOHXqFCkpKSxfvtygySGd0d144+wv49mzZ4LfjFwuJz8/n7y8PExMTPDz8+Px48dIpVK2bt0q\nLD66tpCu4qIT2qpUKtRqtd7ItkgkwszMjCtXrmBmZia0SHQkCZ4HdMpkMmQyGebm5rS0tHD9+nUS\nEhKE7aSnp9Pa2go8J0ODg4OUlZWxePFiMjMzkcvlSCQSTExMcHNzo6WlZcTn7unTpxw7dozFixeP\nK8wPCQmhra2NU6dOsWvXrlFF7oaMpZeVlY3Yj0KhYGBggPnz5ws/CwgIYPfu3Zw4cYLW1laWL18+\nZmVJN631IuEZHh4mLS2NhoYGdu/ejZOTEw0NDROerBOJRKxcuZLk5GROnTrFxx9/TFVVFXV1dfz6\n179GrVbT09Ojp6tRq9VUVVXpER54LlLfvn07J06coL+/f0xLifFgJDzvHoyEx4h3AhOd1KqsrOT+\n/ft89tlnE2pRvAi1Ws2hQ4cwNzdn165dlJaWcuPGDezt7YmPj59QNo+hiI6O5sKFC8yZM2dS3Xef\nPHnCnDlziIyMJC8vj8OHD+Pj40NsbKywcGZkZDAwMMCGDRsQiURERERw+vRplixZMulu1CEhIbi4\nuDAwMMD+/fuJi4t7pdOxTCbDy8uLqqoqAgMDR1RTXn4oFApaW1u5ePEiPT09DA4OIpFIhAVt27Zt\nDA4O4uHhgb29vWByqIO5uTmRkZG4uLhgbm4uPExMTDhz5gwAixYtQi6X09/fT19fH/39/fT399Pc\n3ExXVxcqlYrTp08L1SErKytkMhktLS0UFxdjZ2eHtbU1ISEh5OTkCERDN/H03nvvMX36dIqLi/H3\n98fX15f+/n4OHjzIwoULWbhwISKRiMbGRk6dOsXKlSsNGp9esGABbW1tpKamjhoAC+NPaQ0MDNDU\n1MSWLVv0fl5UVIRUKh0xlefo6EhCQgKJiYn89NNPxMfHj9qOCg4O5tKlS8I0VG9vL6dPn8bGxoaE\nhASB6OoI6UQhEomIi4vj7NmznDx5kpaWFrZu3Spsd+/evSQmJlJTU8Pg4CAmJiY8e/Zs1G1NmTKF\n3bt389NPP9HX1zfhvxOjaPndg5HwGPFOYCIVnq6uLlJSUvj444/fKIPq2rVrwlTNX//6VyHpfLIz\ndV6El5cXJiYm1NTUjNrSeB28OI4uk8lYtGiR0LI6ceIErq6uTJkyhdLSUhISEoQ7fjc3NyQSCU1N\nTZNK7nSCWVtbWzw9PfHy8iIxMRFnZ2f8/f1Rq9UCYXmZxAwMDFBdXS1oVl5+vOix0tfXh1qtpqur\ni/DwcGbNmoWDgwOPHz+msLCQlpYW/vmf/1lv4U1NTaWkpASNRoO5uTmXL1/GwcGBxYsXExERgVqt\n5uzZs0gkEjZt2jSuiP7cuXN4e3sTFhamR4j6+vqQyWQ8ePAAjUZDX18fPT09elUVmUxGVlYWz549\nw9/fnz179mBtbU1vby/Hjx9Ho9Fw48YN8vLyiI2NJSsri/Xr1xvsRq5b+A8fPkxOTs640Q+joby8\nnKCgoBFDAPfu3Ruz4mlubs7HH3/M9evXOXjwoGBS+CJ0FgS6VuDZs2eZM2cOsbGxemRCoVC89o2M\nWCxm3bp1/PnPf8bOzm6EyWNbWxuhoaFIpVKWL18+7rbs7e3Zs2cPJ06cICUlhY8++sjgwQpjhefd\ng5HwGPFOwMrKyqAxZt0d9aJFiya8SKtUKiQSCWKxmOrqavLz84U7MIlEwq5du177S9ZQiEQiYUR9\nsgjPaOPoOn1FdHQ0mZmZZGVl4erqSktLC/7+/ohEIqHKU1paqncttVotSqXylRWW8SovuoWyubkZ\nR0dH3N3d6e7uJisri6CgINzd3UclNFqtlv/6r//id7/73ahu0RqNhsrKSu7evUtvby+Wlpb83d/9\nnd5zy8vLsbOzE1pQL2LVqlV0dnZiZWXFxo0bhaDKc+fOceHCBWG0evPmzeMubFqtlpqaGpYsWSK0\ns17UePX19WFpaYmVlRU3btwgKCiImJgYjh07hkajYXh4GLlcDjz/XGZnZ9PX10dfX5/efuRyOenp\n6fj7+9PT00NlZaVQSbKyshr382pqasqWLVs4ePAgzs7OemTpVS2t0tLSERqhvr4+Ojs7x00WF4vF\nLFu2DFdXV8Gk8OV8qrCwMG7cuMHQ0BDr1q0blcQNDQ29UdUxOzsbV1dXNBoNly5dYtWqVYhEItrb\n2xkaGmJwcNDg7D5LS0s++eQTzp49y6lTp9i0adOYTuYvHr9SqWRwcJD29nah8mfE24WR8BjxTsDK\nyorGxsZxn6PVarl06RLOzs5ER0dPaPv/+I//yJ///GdMTEw4dOgQ1dXVaDQaTExMEIvFdHV10dzc\n/LO0sV7GtGnTuH79up7+5E3w8jj6i4Slra2N4uJiYmNjkcvlJCUlCbEOlpaW9PT00NDQQF1dnZ5Z\nnFQqHZWQ6B42NjYjWkEvVmHEYjH37t2jtraW9evXC8daW1srCIGXL18+6iLg4eExImxSpVJx7949\nsrOzsbCw4L333kMikZCfn6+3+CgUCmpqanB1dR1VJyUWi/nkk0+EJG4nJyd27NjB0NAQBw4coLu7\nm66uLk6ePMnKlSvHnMhra2vD1NR01PdPq9UikUjIysrCzc1Nrz26adMmTp48KUxm6UhnbGys8Prv\nv/+eJ0+eCL+XSCTU1tbS19eHjY2N0FbTRWm82E7T/ftFEfb69es5d+4cu3fv1vMkGotQ9Pb20t7e\nPoKQFxcX6xkujofp06fj6OjI6dOnaWtrY8GCBYhEIlQqFZWVlXR1dfHpp5+OGmILz8X1r2NRAc8j\nIvLz8/n1r3+NqakpR44c4dq1a3zwwQeUlpYSFhZGVVUVixYtMnibpqambN68mdTUVI4cOcLWrVvH\nnCDr6Ojgm2++QSKRoNFo+O677wgLC2Pjxo2vdT5GTB6MhMeIdwKGaHiKiopoamoiISFhQnd/xcXF\nfPPNNwwPDzM8PMyePXtIS0vDyclJqPiIxeJJFe+OB1NTU2bMmEF+fv6od8sTrbDU1dVhbm7Ol19+\nqUdYzMzMkMvl2NjY0Nrairm5OeHh4QwMDNDY2IhGoyE0NJRnz54RHR2Nn5+fHmF5Uzg4OFBQUKD3\nM19fX/bt28f169fZv38/q1atGlEB0E1r6VpF+fn5FBQU4OXlxbp16/D09EQkEpGfnz/C/6WiogJ3\nd3daWloICgoa9bh0JEIHjUbDxYsXcXBwYN++fZSVlXHjxg2++uornJycWLp06YhjfHEc/UU0NjZy\n7do1BgYGEIlE7Ny5U+9aBgQECCaBurH9lxf24OBgmpubhQiNTZs2kZubS11dHWKxmM2bN2NnZ4dG\no2FgYGBEO629vV0gSDpyJBaL2b9/Px4eHtja2grZdWKxWI8wmZmZUVZWRnBwsJ7YWavVUlxcjEql\nMtjPaOrUqSQkJHDq1ClaW1t5//33OXfuHE5OToSEhNDS0jIm4XndltbQ0BDnzp1jzZo1gqh827Zt\n/PDDD8K5rV69msLCwlcadL4MiURCXFwcmZmZHDp0iG3btmFvby9U7HTE29HREVdXV0GkbmJiMuEb\nNCN+HhgJjxHvBF7lttzc3ExGRga7d+9+ZTn5ZfT09Oh9eapUKubOnfuzlZgNJSwPHjygs7NzhJZl\nvAqLzulWV2GRSCRUV1ezfv16rK2t9bxaTp48iY2NDatXrx5BELVaLbW1tWRlZTE4OEhJSQnTp0+f\nVOPGsdyWTU1NWblyJWFhYaSkpFBWVsaKFSsEPVZISAhXr14lOTmZiooKwsPD2bNnzwjH5NEqZGVl\nZVhaWhIaGmpQxpNGoyEpKQm5XM6WLVswMTFhxowZzJgxg9bWVi5fvszp06cxMzMjOjpayJSqqanR\nG8Hu6Ojg+vXrNDc3s3jxYqZPn85f//pXuru7Rxy3lZUVg4ODbNq0iaKiIr2FV6vVMjg4iI2NDbt2\n7eLp06ckJSWxZ88eamtruXz5Mvv372ft2rWEhYUJ1ZzxJgl127x06RLd3d34+PjQ2tqKXC6noqJC\njyzpromDgwNnz54VqkaDg4Oo1Wrs7e1RKBTIZDKDbjqsra3ZtWsXJ06c4LvvviM2Npb333+fiooK\n8vLymD179qiv07WfJwKtVktqaiqBgYF6NhUWFhbs2LGDAwcOCDloLi4ur0WoRCIRS5YswcrKisOH\nD7NlyxZu3LjBwMAAe/fuFZ6zatUqjhw5IlwzQ9tnRvy8MBIeI94JjCdaHhwcFDwxJnpXBvDee+8R\nHBxMVVUVGo2GXbt2jUt2JlJhGU14OzQ0NC5hMTc3x9nZGUdHR6ytrQW35Bcfhn4ZV1RUCMLgF3H5\n8mWGh4dZuXLlmBM6uryqhw8fkpiYyJdffsm8efOIjo6elABKS0tL1Gr1mPlE3t7efP7552RmZvLd\nd9+xfPlyrKysyMnJYXh4GJVKxRdffDGmML2np0fPOFEul9PQ0ICNjc0r/Yjg+cKekpJCf38/H3/8\n8Qiy5+rqys6dO1EqlaSnp5OTk8OdO3fw9/enoaGBTZs20dfXx40bN6ioqOC9995jw4YNwnY8PT1p\nbGwcQXiys7Px9/cnLCxMr22n1WpJS0ujsbGRhIQELC0tsbW1pbu7mxMnTrBnzx68vb05efIk58+f\np6qqitWrV7+SpIpEIiwsLFi/fj0//fQTHR0dODs7ExQUNML1uLW1lR9//JEVK1boVY/KyspQq9X0\n9/fz1VdfoVKp9FpoY7XTZDIZubm5tLa2EhUVRVFREX5+fgQGBpKSkkJfX9+oXkEKhWLChOfevXu0\nt7fr2QHooJuUKykpobCw8I0NN+fMmYOlpSWHDh0Cnl/jzs5O4b328vLCzc2NpqamN7LNMGJyYSQ8\nRrwTMDc3Fxa5F7/AtVot586dIyQkxGADsNEIy6FDh/jhhx8IDg4mODiYK1eujEtYTExMRpAQmUwm\nEBZra2ucnZ3HJDWGfFk7OzuTkZEhCCpfB6OloxcWFlJdXc2nn35q0HGEhobi7u5OREQET5484csv\nv2T27NnMmzfPYKfb0SASiXB0dKSrq2vM1oVUKuWDDz7A3Nyc5ORkRCIRCxcuxNvbm6dPn447hfdy\nhefhw4dMnTqVjo6OEb4qL0MXNtvT0zMq2XkRpqamfPTRR6xatYri4mIyMzNRqVR89dVXDA8PM3v2\nbL744osRpM7T05OGhgY9N2O5XE5eXp5QDdBBo9GQnJzMs2fPRjgFz507l87OTs6ePcvWrVv5/PPP\nuXbtGgUFBdTV1bF161aD2kwSiYT4+HgOHDgw5nWtqqoiPDxcr13X399PUVER/v7+BAQEEBkZiUql\nEtplL1aIGhsbhX/39fUhl8sRi8U4OTnR09ODq6srx44dIzg4GFdXV+7evcu8efOwsrLS+6xOtMLT\n2dnJ1atX2bVr16jvpW6Sce3atZw/f35C5pZjQZfCPjw8LLRYV6xYIfz+ww8/5MyZM5NuimrE68NI\neIx4J5CTk8PXX39NTU0NBw8eZHh4mKGhIbKysujt7SUqKoqSkhKDKywvExaVSoWtrS2mpqZoNJpJ\nISxvioCAANLS0njy5Amenp4Tfv1o6ei1tbVkZmaye/fuCbXsIiIiaGpqYsOGDXR1dXHnzh2++uor\nZs6cSUxMzIQjP3TQpaaPRniUSiXFxcXk5ORgbW3NunXraGlpIScnhwULFlBZWcnw8PCYramXCU9p\naSlmZmZMmzZtXAKpa310dXWxbds2g1ukYrGYGTNmUFZWRmNjIyKRCKVSyb179zAzMyM2NlavMufp\n6TlCw5SVlUV4eLheDMrw8DBnz55FrVazffv2EQu2zlDvxIkTXLp0idWrV7N8+XKCgoI4c+YMBw4c\nYNmyZURHR7+SOFtYWPDxxx/zt7/9je7u7hG/LysrY+XKlXo/KykpEaIeFi5cCDwnqvb29mPGueiE\n34GBgSxcuJChoSGBGDk4OHD//n1MTExobm6mrKyMgYEBISndysqKjo4OhoeHhc/Gi5Wkl98vtVpN\nYmIi77///pjET6dZCg0NJSMjg/LyckJDQw0e838ZWq2Whw8fotVqhfHzgoICli1bRnV1NX/+8585\nfvw4/f39/I//8T/YsGED//AP/zBulIcRPz+MhMeIdwK///3vaW1t5ciRI1hZWeHm5oaJiQkKhQI3\nNzeKi4v1SMlECMvQ0BB/+ctfgOel8iVLlryt09TDiyPqr0N4Xh5H7+zsJDExkU2bNk04ITwsLIyM\njAxUKhUODg6sWbOGRYsWkZ2dzf79+wkLC2P+/PkTzhEbTcfT399Pbm4uRUVFeHt7s2HDBuH8w8PD\nCQsLE6o9paWloy4SuhRwKysr4PnI9NOnTzExMRn3/dVqtVy8eJH29vYJkR2tVsuDBw/IzMxkcHCQ\n1atXM3PmTIaGhrhy5YqQZB8SEsKKFSuwtrbG1dVVMETUmewVFxfzm9/8RtiuQqHg1KlTWFhYEB8f\nPybRFovFbNq0iUOHDpGbm8u8efPw9fXlt7/9LefOnSM9PZ3Kykri4+NfGVvi4uKCu7s7OTk5REZG\nCi2l9vZ25HK5XntUq9VSVFTE6tWrKSsrMyhepaqqiuTkZCFqYjQStmTJEsEAUBfUKZfLBVKUnp6O\nVqulq6uLhoYGvUqSWCzWI0GdnZ2oVCrMzMyoqakRfmdubi7su6ysjPDwcMFteevWrZw9e5Zf/epX\nr6wGjgaRSMTnn39OT08Pjx8/5sGDB9TX1/OXv/yFf/mXf0GpVAoePIODg0Ib8n/9r//F3//93094\nf0ZMDiR/+MMf/vC2D8IIIxQKBRcvXsTHx4fExERmzZpFcXExW7duZcmSJUybNo3Q0FACAwPx9fXF\n09OTKVOm4OzsjL29vaAXkEqlI/QvqampPH36VDDECw8Pf6NWzWTCycmJtLQ0Zs6cOWExdklJCTKZ\njODgYAYHB/nxxx9ZtGgRoaGhEz4OU1NTamtrMTMzExY1MzMzAgICiIqKor29XbiOjo6OAtF4FXp7\ne2lubiY0NJT29nauX7/OpUuXcHBwIC4ujtmzZ4+YtLK2tiYqKkoYL7a2tsbNzU1v4ezq6qK6uloY\nPb93756QTD3WuLFOI/P06VO2b99ukE5Jq9Xy+PFjzpw5w9OnT1myZAkVFRWsX78esViMiYkJISEh\nxMbGIpPJKC0t5ebNm1RWVuLk5ER3dze2trY4OjqSnp6Op6en8P7I5XKOHTuGk5MTa9eufWVV0cTE\nRNC+ODg44OTkhFQqZdq0aVhZWVFYWEhBQQGenp6vTC+vrq7G3t6ewsJCpk+fjlgsJi8vD0dHR72q\nR319PTU1NQQGBtLR0TFupIVWq+XWrVvcvHmTzZs3ExoaOmbFycTEhIiICB4+fEheXh6BgYHY29tj\nZWWFg4PLIZMsAAAgAElEQVQDpaWlAgkMDw8nMjKSefPmsWDBAqKjowkMDMTd3Z2BgQFqamoICgqi\nvb2d6upqioqKuH37Nrdu3aK4uJjS0lIePXqEpaUl9fX19PT0EBgYyJQpU0hNTcXHx2dCFUylUklK\nSgpOTk44ODgwZcoUZs6ciYmJCbt372ZwcFCwPXjx2qhUKm7fvk1ERMSIqT8jfhkYCY8Rbx11dXXk\n5OSwaNEiDh06hIWFBceOHWPWrFkjRJUThc6VWeeJofNImSzTvzeFVCoVvF8m6vCcmZnJ9OnTsbe3\n5+TJk3h7e7NgwYLXPhaNRiNMRb18jL6+vsyePZve3l4uXbpEQ0MDzc3NrFixgn/7t39jypQpo+oi\nlEolBQUFVFdXk5WVhZ+fH+vXryc8PHzclptYLMbHx4fCwkK6urqoqKjAy8tLqF40NzfT2dkp7PPK\nlStIJBJCQ0NHrZZptVquXLlCU1MT27dvNyi8tbm5maSkJCHzatmyZbS2tjI0NDQi/FIkEuHh4UFM\nTAz+/v48fvyYrKwsBgYG6OnpwcPDg6tXr7Jp0yakUil9fX0cOXIEPz8/VqxYYbBI3dzcHC8vL86e\nPYufnx/W1taIRCJBg6UjEEqlEh8fnzEJh66l09fXx+PHjwkKCuLSpUssWrRIb/G/fv06wcHB9Pb2\nIpVKCQwMHHV7Q0NDJCYm0tbWxo4dOwyqBIlEIuzt7amrq6OoqAhHR0ehWllQUICZmdmIz5RIJMLE\nxARLS0tMTU25dOkSGzduZO7cuYSFhTFz5kzBuXnevHmEhoZibm5OU1MT4eHhtLS0oFAo6OzspK6u\njsHBQfLz88nPz6esrEzI3mppaaGjo4O+vj4UCgWA4NnV3t7O5cuXKSoqQqlU4uXlhVgsZufOnVRX\nV497ziqVipKSEv7bf/tvr7w+Rkw+jC0tI94aNBoNmZmZZGdno1arkUgkWFhYcOXKFWxtbSdshz8a\ndNbw5eXl1NbW4u/v/8YTGpONOXPmcOzYMWJjYw3WDikUCpqamvDx8eHSpUtIpdJxHXANQUhIiCDm\nHo0QmJmZMX/+fObMmcO9e/dYvXo17e3tACQkJLBy5UqhlabRaCgvL+f27dt0dXURExNDfHz8hMbe\nbWxscHZ2ZsGCBbS2tnLgwAHef/99Zs+ezbNnz4QqxrNnz+js7ESj0bBu3boR29FqtaSnp9PQ0MAn\nn3zySrLT1dVFRkYGDQ0NLFq0iMjISIGQjOW/8yI8PT3Zu3cvAwMDnD17lrq6Or799ltBX9Ld3c3R\no0eJiorSMxw0FFOnTmX16tWcPHmSTz/9VCAoDg4O/OY3vyE9PZ3s7Gyqq6vZtm3bmInpIpGItWvX\ncvjwYa5evYpardaLYZDL5Tx69IhVq1Zx8eLFMclOe3s7p06dws/Pb9y23Gjw9fVFoVCwZs0a0tLS\naG1tZeHChQwPD4/7WdFqtSQnJzN9+vQx3w9TU1McHBzIz88XKkRdXV34+fnpfbeUlJRw7do1QYOl\na501NzfrCbPlcjnm5uaYmpqi1WoZHh4mOzubgoIC5s6dS3Z2tkHnXF9fz4MHD974Zs6IicNIeIx4\nK5DL5Rw9epTOzk7UajUmJiZoNBoePnxITU0Ne/funZRAS5FIhKenJ83NzXh5eb0xKfg54OrqioOD\nA5WVlQZPotXW1jJ16lSKi4t58uQJe/bseWOzQJlMho+PD5WVlSMqGC9CKpUSHR2tt7CJRCIGBwdR\nKBSCENnOzo7FixeTlJQkZBdNFGFhYVRUVBAXF0dISAjJycmUl5fj4OAgCJZLS0sFH5qXF3etVsu1\na9eoq6t7JdkZGBjg5s2blJaWMm/ePOLi4vTajLo4CUOJuKWlJZs3b+aPf/yjsJD+x3/8B2KxmLlz\n574W2dEhLCyMrq4uTpw4oedNJZFIWLlyJaGhoZw8eZKvvvqKDRs2jGih6KIldA7C3377LYGBgXp/\ncyUlJQQHByOTyWhubh61VVheXs7FixdZtmzZawlyJRIJISEhdHR0kJCQIDgzv4rw5OXlMTAw8MqR\nb61WS1lZGTt27ACeJ86/3PKdMWMGw8PDXL16ld27d4/Z3tJoNMjlcgoKCsjKykKtVgvu0UVFRZib\nmwvVoPFgYmJCbW2tkfC8Bfy8wUFGGDEGFAqFXq9bp4dIT09n8+bNBrUcJoK+vj6DdSdvA9HR0eTl\n5Rn8/EePHmFnZ8edO3f4+OOPJ8U3B56LhktLSw167p///GdBIL548WKSk5P5y1/+wpMnT4iPj2fX\nrl2EhIQIo+mvg9DQUCorK9FoNDg7O7Nnzx4CAwMpKSmho6NDWNBUKtWIBUSr1ZKRkcHjx4/ZsWPH\nmC00hULBjRs3+OabbxCLxXzxxRcsXLhwhKaqq6sLjUYzIS8onadSVFQUW7duFQT32dnZ/OlPfyI3\nN3eE3sNQzJ8/Hzc3N86dOzdiGz4+Pvz93/89Hh4enDlzhvPnz49I7taRG930Yk1NjeD+rBMrR0VF\nMTg4yMDAgJ4QXqPRcP36ddL/P/bOOyiqPN3fTwcyIiIZyQiIgBFwdEwYMGDA7Jid0Vl1Z3Znt+7O\nra2turdu3d/d3do0u6vj7KhjHAOKigFH0QFEQQUMRMlKBpEcGujw+8Pqc23phkZlnL3bT5V/SB9O\nn3M43d/3vO/nfT/XrrFu3bo36j4KDAwkJyeHIUOGsGnTJoyNjXU6mMOLWUE3b95k+fLl/WaTysvL\nBZ8zpVJJbW2t1gzvhAkTCAkJ4ciRIzoHoKqnUhsbG6NQKHBzc2PZsmX8+te/Zvbs2Xr/HUUi0Vv7\nvBoYGIaAx8A7YdiwYezatQtjY2PMzMyEroaIiAi9R9cPhLa2Np2p/R8D/v7+NDQ0COPo+0KlUlFQ\nUEBeXh6rVq16K35cavz8/CgvLxeMLftizZo1ZGVl8fnnnxMWFkZ9fT0ymYyqqiqePn0qTO5Vt6a/\nDtbW1lhbW/PkyRPgxaIzefJkhg8fLpS5mpubqa2t7fXknpiYSEFBARs3btQqUlcoFKSlpbF7924a\nGhrYtm0b8+bN0yloLykpwcvLa0CZx/LyckGwevz4caKiovj5z3/OL37xC1xdXbl27Rq//e1viY2N\nRSaT6X9heLFwRkZG0tXVRXx8fK/XTU1N2bhxI/PmzSMnJ4e//e1vWv8OlZWVmJmZMWvWLE6ePIlM\nJhOO283NjerqapycnIQMYmdnJ8ePH6eiooJt27b1ciMfKO7u7oIxqVQqZfHixRgZGfH48WOePn2q\nsW1PTw9nzpxh7ty5enUMZmdnC5q0hoYGzM3NdQa+kydPZvTo0Rw9epTOzk6d+5wwYQK/+MUv2LJl\nCwEBAUgkEgICAvTOsHZ1dWmMkjDww2EQLRt4Zzx48ACZTMb27duFmRbLli0blPdKT0/Hy8trwO3a\nPxRisZju7m5KS0t1ekCpUXcvRUZG9rvtQJFIJNTW1tLT06NzIVPbUsTFxQlTax0dHYUsjkwmo6Sk\nhPr6eoKCgqitraWzsxNPT8/XOib1AvzyuSYlJbF582ZKS0upr69n+PDhhIaGCsFIUlISubm5bNq0\nqdeQPZVKRW5uLtHR0bS3t7NkyRJCQ0P7nVukdnrX13NNpVJx7tw5hg4dSn5+PmvXrhXE8iYmJowe\nPVoQmWdmZpKYmCgYn+obnIvFYvz8/Lh+/ToikUjrvCMXFxcCAwPJysri1q1bmJqa0tTUhIODA3Z2\ndqSkpODs7MzkyZOpr6/nwYMHPHv2jJEjR+Lm5kZOTo7QsVdTU8ORI0fw8PAgKirqrWQqRCIRTU1N\nNDc34+7ujkgkIjU1FT8/P27duoWJiYlwL165cgVzc3NmzJjRb+CpVCq5ePEic+fOxdzcnJKSEmQy\nWS9R/st4eHjw/PlzUlJSCAwM1JpBkkqlvc5bIpHQ3NxMWlpar0zayxgZGbFmzRrWrFnT57EbGBwM\nGR4D7wS5XE5ycjLh4eHcvXsXlUo1qGLiH3uGB148Oebk5PT5pC+Xyzl37hwODg596mzeBF1lLYVC\nQWZmJl9//TVXrlwhICCAn/3sZ4SFhQnTZtWoPYeANyppwYuy1uPHj4WSQU9PDzKZDEtLSxobGxk2\nbBgqlYpvvvmGuro6bt68SXZ2Nhs3buwV7JSWlrJ//35u377NwoULWb9+fZ8+VGqUSiVPnjzpV7D8\nMuoSUU1NDaamplq78MRiMdOnT+ff/u3fWLduHV1dXezbt48vvviCjIwMvcokZmZmfPDBByQlJens\nErKxseGTTz5h3LhxXL16lbKyMuRyOUqlkpycHAIDAwGIiIhAJpORk5Mj3F9VVVU4OzuTlZXF0aNH\nmTVrFnPnzn0rBrNqRo8eTU5OjvB/hUKBo6MjW7Zs4c6dO1y+fJmcnByKi4v1nkxeVlaGpaWl8JBT\nXV3d73eMSCQiIiICOzs7Tp48KczS0Ydf//rXeHh46NQeSaVS7Ozs+MMf/qD3Pg28XQwBj4F3Qlpa\nGk5OTvT09JCamkpISMigmXnCj1/DAy9MJUeOHMnDhw+1vq62Q+jp6dE5a+Zt4OPjQ21tLS0tLcCL\nFHxKSgp/+9vfePDgAeHh4ezcuZNx48YhlUpxd3enurpa42nY2NhYKM/pMhHVFxsbG4YMGUJZWRnw\nwkNr6NCh1NXV0dPTQ0dHB9u2bWPs2LHs37+fO3fusG7dOo2/d01NDd9++y0XL17kvffeY9u2bQMK\nXqqqqrCystL7HlKpVFy4cAGFQsHmzZtRqVQ0Nzf3+TtqJ/Wf//znODo6EhcXx+9+9zsuXbokzBjS\nhY2NDatWreLcuXPU1dVp3UYsFhMZGcn69evp7Ozk0qVLpKWlYWlpKeiSJBIJvr6+iMViiouLgRcl\nr+LiYhISEti4caMQHL1N3Nzc6OzsFLr+lEolJiYmDB8+nA8//JD6+nrOnj3LwoUL9db3vVzOghf3\ngD4PVepSobm5uTABWx+GDBnC1atX8fLywsTERNCAqX31Jk2aREZGxqCU7A3oh6FLy8APTnd3N7dv\n32bZsmXExMQQFRVFQ0PDWxcqq1EoFMhksj59mX4shISEcP78ecLCwno9xd66dUvoYBnIYj1Q1MP0\nMjIy6Onp4eHDh3h7e7N69WqtZS4rKytMTU0ZPnw4TU1NhIeHc+nSJaHV2d/fn+fPn6NSqV678y4g\nIIDc3Fw8PDwES4ns7Gysra1xcHBAKpXS3d2Nubk51tbWnDp1iiVLlmBqakpCQgLFxcVMmzaNNWvW\nvJZtiD7t6GrU/m/t7e3s2rWLYcOGCUai/Q0EhBci4jVr1iCXy7l58yZpaWncv38fNzc35s+fr7Ok\n5ubmRkREBCdOnODDDz/UGZx5eXnh7e1NU1MT3333Ha6ursLfRqVSkZ2dTUREBFevXtVo0962bdug\nPZSIRCICAgLIzs5m5syZQsADCG3gI0aM4PLly6xZs6bfsqK641NtJKpSqaipqdErmwcvgsOoqChO\nnTpFbGwsS5cu7TejpVKpuH37Nv/4xz8YMmQIhw4dwtjYGFtbW5YvX66zrd/AD4chw2PgB+fu3bt4\neHiQlJRESEgI3t7eOh213wbt7e1YWFi8lTb3wWbEiBGYmJgIT9dqcnNzSU9PJzQ0FBcXlwFPZR4I\nNTU1NDc3k5ycjFKpZPv27SxfvrxPcaqbmxv+/v789Kc/JSAggNmzZ2NkZERiYiK5ubmIRCK9hNC6\nCAgIEHRe6hk82dnZNDc3ExwcLMxD2bp1K5s2bWLs2LEcOHCAPXv2YGVlxSeffEJoaOhre6SVlJTo\nNaxSPeAwLy+PRYsWCV5T6oBnIKhtMj7//HNWrlxJe3s7X331FX/961959OiR1t8JDg5mzJgxnDx5\nkp6eHp37VnfWGRkZUVFRwZdffkl7eztVVVV0d3czfvx4Jk+ezNmzZ7G0tOSDDz4Y1Aws/G+3lkql\nQqVSCQHP7du3Adi0aRPh4eEcOXKEvLy8PvdVWlqKtbW1cP1bW1sRiUQDyvKqzVZbW1u5fPmy0Mqv\ni5ycHJqamggLC+PGjRs4Ojryxz/+kX//9383BDs/EgwBj4EfFJlMxp07dwThn1q0qWvY3dvgn6Gc\npUbtr/Vyi3pVVZXwZFtRUTEoX55qC4WjR49y/PhxPDw8MDMzIzQ0VK8uMDc3N6HkBDB+/Hj8/Pyw\ntrYmOTkZExOTNyprDR8+HHNzc8rLy2lqakIsFiMSiRCLxVRWVnLv3j02bdqEmZkZt27dIikpiYCA\nANzc3CgoKHjtLjF4UdKrqanp13NJqVRy4cIFCgoKek2efp2A52VGjRrFrl27+OSTT7C1tSU2Npbf\n/va3XLlypVe5a/r06djY2HD+/Pk+F+na2lrs7e3ZtWsXXV1d/OUvf+H69euMHz+ehw8fkpKSgp2d\nHSqV6rVb5weCi4sLcrmc2tpaVCoVpqamVFRUcPfuXcHKIygoiHXr1vHdd9+RlJSk8/zU3llq1Pqd\ngT70qEXGtbW1xMfH63y/zs5Orl69SmRkJLGxsXR2dqJQKGhraxvQ+xkYXAxdWgZ+UG7evIlCoaC2\ntlbDGTonJwcbG5s3bnHVRnV1Nc+fP/+nGfRla2tLfHw8o0aNoqenh6NHj7JgwQJhqvKMGTPeWnlO\noVCQlZXFuXPnKC4uZsKECSxevBhPT0+am5tpamrSy1zR2NiY5ORkjXZbLy8v8vLysLOzo7q6GplM\npvdgRW10dHRQXV1NS0uLYCJpZWVFQUEBGzZsoKSkhOjoaCQSCcuXL2f8+PEEBwdjYmLC+fPnBWPM\ngYptS0pKaG5u7tNHSi6XExMTQ3t7Ox0dHSxcuFAjULSwsCA+Pp6wsLDXzjLBC4FycHAwU6ZMEdzm\nk5KSKCsrw8XFBXNzc0QiESNHjiQtLY3GxkatpbisrCwaGhoICAjA19dXGCugDg4rKirYsGEDVVVV\nQlDp5+c3qFlSkUhEa2sr9fX1lJWVMXHiRM6cOcP8+fMZMWKEsN2QIUMIDAwkOTmZoqIiRo4cqXFN\nFQoFFy9eZP78+cJDVE5ODubm5q9VCpZKpYwaNYqEhAQ6Ojq0is/j4uJwcHAQBm+qh6na29vr3dVn\nYPAxBDwGfjDa29s5d+4cHR0drF27VmOOxqNHj3B2dh4UQd+TJ0/o6urCz8/vre97MJBIJHR0dFBR\nUcHt27cZM2YMEyZM4NmzZ2RnZxMeHv7GC49MJuPu3bvExMTQ1tbG9OnTmTt3rsa8FRMTE27dukVI\nSEi/+1NnVgIDA4VFRiQS4evry82bN7G1tRVmu7yOOzW8CBouXLhATU2NMOpfJpMxdepUrly5QlNT\nE4sXL2bSpEnCLB2RSISDgwPBwcFkZmZy8+ZNnJyc9NLSqLl37x4ODg46j7u7u5tTp04hkUjw8vKi\nq6url6eZWCymoKAAW1tboczyJojFYry9vXn//fcZPnw4jx8/5ubNm2RlZWFpaYmjo6Pgj/XkyRMC\nAwM17pnMzEwqKipYuHAhJiYmiEQiGhsbKS8vRyaTIRKJGDt2LLdv32blypVCu7VMJuPs2bOMHTt2\nUIIfMzMzkpKSkMlkdHZ24uDgwJQpU3ptp/bZKikpITk5GR8fH+G+Ky4u5tmzZxoTse/cuYO3t/dr\nf78YGRkxatQorl69KuiJ1JSWlpKamsqKFSs4efIkIpEIpVKJUqkUgiUDPw4MAY+BH4zr169TV1fH\nnDlzes2PycjIwNvb+60sBq9SUFCAsbHxa8+BeRcMGzaM2NhY3N3diYiIQCQSabijvy5NTU0kJSUR\nGxuLqakpCxYsYMqUKdjY2PRawKysrLh9+zZeXl79lgRFIhEVFRUYGRlpPNFKpVJ8fHxISkrCzs6O\niooKOjo6hHkrA+Hrr7/mN7/5DSkpKSQnJ+Pp6Ym7uzuVlZXMnTuX8PBwnbYAxsbGjB49GgsLC2Jj\nY2ltbcXd3V2vbMu1a9eYPHmy1rEG6iF81tbWLFy4kJiYGBYvXqx12/r6euHc3yb29vaEhYUxevRo\nysvLSU1NFfzpenp6KCkpoaWlReO+uXv3LkZGRoK9RVlZGefPn2f06NFs2rSJ/Px8EhMTUSgUzJs3\nD09PT6Kjo8nKyqK9vR1/f/9BKRNbWlpy9+5dZDIZKpWK1atX6/wbqWcQKRQKYmNjcXZ2FkqoXl5e\nGkHJ9evXmTJlis6hkvpgbGyMv78/ly5dwtjYWOgyPX78OPPmzcPJyYmwsDDBENjR0RELC4t/mget\nfwUMXVoGfhCam5u5f/8+/v7+WksDnZ2dg6bhaWtr+6dLKz98+BAjIyNcXV2FwKCoqIiwsLDX2l91\ndTUpKSkUFxczduxYfvKTn/Sb5RCJRAQGBpKdna3X9VPreF51uLaxsWHWrFl89913bN68mYsXL6JU\nKpk5c6beQU92dja/+c1vkMvlwmyU48ePs2PHDoKCgvQuU40ePRpPT0+uXLnCV199xeLFi/sMQFpa\nWmhvb9faztzW1saxY8fw8PAgIiKC27dv4+bmprMTyNXVlfv37+t1nK+DnZ0dGzdupLu7mxs3bpCW\nliYIlx88eICjoyOhoaHCefn5+aFSqUhPT+f777/H1NSUJUuWIBaL2b59O2fPniUrK4sDBw7Q3t4u\ntGdLpVLKy8sH5TMlEolwd3enqalJcJbvb/tJkyZhZ2fHmTNnmDp1Kvn5+cyaNUvYpqOjg87OTr0m\nM/fH0KFD2bBhA4cPHxZGLzg6OgpBjYmJCT09PQQGBmocg4EfBwbRsoEfhJiYGOELVRuDKVr+Zxg6\n+DKZmZlkZWURGRlJRkYGKpVKcEcfSJZKpVJRWFjIkSNHOHnyJE5OTnz66afMnTtX75LOy50z/eHu\n7q4hXH51PyKRiNjYWFavXk1BQQHXr1/Xa78A+fn5SKWaz2dtbW34+/sPWJNjbm7O8uXLmTNnDjEx\nMcTFxemcc1NSUoKnp2ev92hqauLgwYP4+/sTERFBV1cXqampfZpZurq6UlFRofc5vy7GxsbMnz+f\nkJAQjYDyypUr3Lp1i127dvHrX/+ajz76iN///vekp6fj4+NDaGioxnkOGzaMcePG8ezZMw1vK7lc\nrnPA4ZuiUCgEcfdAAipvb2+2bt3K7du3kUqlGpkc9fydt1WCGz58OOvWrSMuLo60tDTmz5+v8Xpl\nZaVGdsnAjwdDwGNg0MnNzaW8vJwPPvhA5xOboUvrBeXl5Vy9epW1a9cSEBCAXC6nvLyc0tJSRowY\noVc7ulwu58GDB+zdu5cbN24wduxYPv30UyZPnjzga+zo6CiIVvXZtrm5WWv7uZmZGcbGxri5uXHl\nyhXWr19PaWmpoInoDz8/v15t1paWlhw/fpysrKwBTcRV4+/vz44dO+ju7mbv3r2UlJT02kbtn/Uy\n9fX1HDx4kJCQEMHiICUlBV9f3z6NRS0tLTEzM6O+vn7Ax/o6VFZWYmxsjKWlJVZWVojFYv785z+z\nf/9+uru7qa2t5b/+679YvXo1RUVFjBs3TuP3q6urcXV11brv4uJiWltb2bt3L2vWrGHLli1cvnxZ\n7yF9ukhISBAE+RUVFQP6XRsbG1xdXTE3N+fo0aOCCai+AwcHgp2dHRYWFqhUKmFYIrx4yKisrNRq\n8WHg3WMoaRkYVNrb2wVtQF/eTF1dXf/yGZ6mpiaio6NZunSpIK5Ut6irvYz6orOzk/T0dEFkq9Ze\nvMmT7ctlrf6eWsViMSNGjKCsrAx/f/9erw8fPpzg4GCSk5NJTk5m48aNHDt2jLi4uH7tAgIDA/ns\ns8/4/e9/L4w0uHr1KlZWVmRkZHD16lXGjBnDxIkTB6QDMzMzY+nSpRQWFhIbG4uPjw9z5szB1NQU\nlUpFSUmJYJEBL4KA48ePM2vWLMEhvK2tjfT0dLZv397v+7m6ulJWVoadnZ3ex/i6bN68WeP/KpWK\nqKgojWyWSqUiIyMDNzc3Df2TSqWiqqqKkJAQhgwZImR4FAoFYrEYsViMs7MzKpVKCCxiYmKwt7cn\nMTHxtTIcpaWlZGZmMm/ePMrLy8nJydEZcGmjp6eH4uJidu7cSVpaGvv372fNmjXU1NS8df3evXv3\nsLCwYMGCBZw5c4a1a9cyYsQInj9/jqmp6T/NA9a/GoaAx8CgoVQqOXHiBCqVisjISJ3bdXV1YWRk\n9Fa9edSoVCra2tp+9FOWu7q6OHHiBFOmTNGYszNmzBgSExORSqU6HZYbGxu5c+cOmZmZ+Pn5sX79\n+reqrwgMDOTw4cN6+SepdTzaAh4bGxsaGxtZsWIFBw4cwN7envXr13P8+HEuXrzIokWLdAY96u6u\nX/3qVzQ1NfE///M/Qtt3QEAAz58/JyMjg/379+Pk5MTEiRMFiwR9GDlyJDt27CA+Pp69e/cSGRnJ\nkCFDMDExEd7n6dOnREdHExkZqdF5k5ycTFBQkF7zitRlrQkTJuh1XG+T8vJy3N3dMTIyoqenB4lE\ngouLC6mpqdjY2HDq1CnBBkGtw/Lx8WHkyJGCNUZ5eTkxMTH893//d68yoLpzbsaMGeTn5w+o/b6j\no4Nz586xZMkSuru7EYvF5ObmCoJ9fSgqKsLZ2RkrKytmzZqFvb09R44cQSqVMnnyZP0vVD80NTVx\n8+ZNPvzwQ4YPH86SJUs4efIkGzZsoLq62lDO+hFjKGkZGDQSEhJoampi2rRpfWZvZDLZoE1x7ejo\nwMTEpJf+48eEUqkkJiYGV1fXXqJkU1NTvL296enp6VUuqays5MyZM+zbtw8jIyN27NjB0qVL37qY\n1NbWFktLS536nJfpS8ej9tQyMzNj7dq1JCQkUFNTw7p162hoaCA2NlbngLu8vDw6OzuxsrLCwcFB\no4wAL7JHc+fO5bPPPiM4OJiUlBS++OILEhMTBU+w/jA1NWXRokUsWbKEuLg4Lly4gJubGwCFhYVE\nR0ezfPlyjWCnqamJrKysXm3ounjTAYSvQ1tbG+fPn+fMmTNERETw4Ycf4uvry9ixYzl79iwAz58/\n5xI8YsUAACAASURBVPHjx2RlZZGenk51dTXOzs5CsCESibC2tiYoKIjq6mqdpSuFQkFdXR3fffed\n3sen9h0LDAzE29ubrq4uxGIxZmZmet1zal4dNhgUFMTq1atpaWkhNzf3rWinVCoVly9fZtKkSYIp\nqa+vL/Pnz+fbb7+luLjYUM76EWMIeAwMCvn5+Tx48ACg386iwe7Q+rGnl+Pj45HL5cyfP1/r06yV\nlRUKhQKlUolKpSI/P59Dhw5x+vRpRowYwc9+9jNmz56tsyX7bRAYGEhWVla/27m4uFBXV6dVBPyy\niejw4cNZtmwZZ86cob29nQ8++ICWlhbOnz/fK+hRKpWCtqOrqwt40SqubQGTSqUEBwezdetW1q1b\nR3t7O3v37uXUqVMUFRXpteh5eXmxY8cO2tvbyc/P59q1a8TGxrJ27dpeep6kpCQmTpyo9z1mZ2dH\nW1vbG9ls6ItSqeTevXvs3bsXc3Nzdu3aRXl5OY6OjmzcuJHFixeTlZVFSEiIhrbO3t6ehoYGnSXo\nc+fO9anVaW1tJTo6Wu/jTE9Pp6WlRehqUgc8rzqo90V3dzdFRUW9Zt6IxWLs7e158uQJp0+f7teE\ntT+ys7NpaWnpNRto9OjRzJw5k9zc3AHNeDLww2IIeAy8dRoaGrhw4QI2NjZMnTq1X6HtYAuWf8z6\nnYyMDAoLC1m5cqXOEkBNTQ3Dhg3j8uXL7NmzR1hkP/30UyZNmiR4Dg0mo0ePJi8vr19RqnoOjzbB\n6fDhwzXsJby8vJg+fTonTpxAqVSydu1aOjo6iImJ0XifzMxMTE1Nqa6uFn7W3NxMfn5+n8fi4ODA\nwoUL+eyzz/Dx8eHGjRv8/e9/5/bt24LuRBdisRiZTEZgYCB37tzBycmpV1uzejLxQMolap3TYGd5\nysvL2bdvH3l5eWzatIm5c+eSk5NDTU0N8CIT4+rqSlFREQEBAURGRiKVSpFKpdjZ2ZGWlkZra6vW\nv3dfHl1qZDKZXsdZV1dHQkICy5cvF+7/rq4uJBIJgYGB5Obm6mVrUVBQwIgRI3rN2VFnqjZu3IiJ\niQnffPONRsfZQOjo6ODatWssWrRI62dV3YkYHx9vsJT4kWIIeAy8VXp6eoiOjmbMmDG0tLTopVUY\nzJLWjznDU1paSkJCAmvXrtV5/k1NTZSVldHc3ExeXh6RkZFs27aNwMDAQdE86cLa2hpbW1utnUyv\n8qqvlhp1huflLEtISAgeHh7ExMQgkUhYs2YNPT09nDlzBoVCgVwuJzExEU9PT5RKpZABUyqVJCUl\n6XXsxsbGTJgwQTBBra+vZ/fu3Zw9e5anT59qzfqUl5djampKYWEhH3/8Mba2tuzdu1cj45CQkMB7\n77034GB9MAOe9vZ2YmNjOX36NJMnT2bjxo3Y29vz6NEjLl68qLGtunPu0qVLjBw5Ej8/P2bOnMmy\nZcuQSCQ0NDSwZ8+eXmMJ+vtMm5ubM3369H6Ptaenh5iYGObMmSOUhwBBw2NjY4OVlRVPnz7td1+v\nlrPUqB3SpVIpixcvZty4cRw4cIAnT570u89XiY+PJyAgQKdGp7q6GgcHB8aMGcPRo0fp7Owc8HsY\nGFwMAY+Bt4ZKpSIuLk7wTpo2bZpe2pnBnsHzYwp4rl+/zsWLF6mrqyMmJobly5drfNmraWhoIC4u\nji+//BIzMzO2bt2KsbGxYAPwLlB3a/WHLh2PqakpUqm019PvvHnzUCgUxMfHI5VKWb16NQCnTp3i\n3r172NvbU1NTg7GxMdOmTcPY2JgVK1b0KYTXhkgkwsXFhSVLlvDpp5/i7OzMpUuX2Lt3rzDdF17c\nxzdu3KCnp4ctW7bg4OBAREQEq1atIjExkejoaIqLiykrKxMG+Q0ENze3tx7wKJVK0tLS+PLLLzE1\nNWXXrl0EBQUJ98qrpSGVSkV1dTUqlQovLy9Onz5NVFQUkydPpqmpCRMTEzZt2kRkZCS3b99m//79\nQpDwq1/9qs/Pq1wu73MekZr4+Hjs7OyEbjc1akE1vMgs9nfPdXV1UVpaqlUo/3JLukgkIiwsjKio\nKM6cOUNaWlq/x6impKSE0tJSjY69V6moqMDFxYVp06bh7e3Nt99+K5RgDfw4MAQ8Bt4aDx48oLKy\nkqCgIJqbmxkzZoxevzeYGp4fW0krKyuLR48esW/fPqZMmdKrXbaiooLo6GgOHDiAiYkJ/v7+TJ48\nGQcHByZOnDigL+m3TUBAAPn5+f2WNFxdXamsrNRaDnm1rAUvvMNWrlxJQUEB9+/fRyKRsGLFCiQS\nCd9//z2hoaE8efIEKysrQkNDkUgk+Pv7v5E41MzMjEmTJrFz504WLFhAeXk5f/3rX4XsSF1dHUuW\nLNHQRbm6uvLxxx9jY2PDiRMn8PLy6ncSsDZcXFz6FP4OlIqKCvbv3092djYbN24kIiKiV5kzMjIS\niUSikRVsb29n6NChBAYGIpVKuXz5stCOrtbveHl5sW3bNiZNmkRsbCzHjx+nsbGR6dOn9yofSSQS\njI2N2bNnD5cvXyYxMVFnOSo/P5+CggIiIyN7BfDd3d0aAc/jx4/7vFb5+fm4ubn1ypIqFAqePXvW\nawaPl5cXW7duJS0tjUuXLvX7d+jp6eHSpUssWLCgz/KxeuCgSCRizpw5ODg4cPLkSb1KgAZ+GAwB\nj4G3QlVVFTdu3GDlypUkJyczY8YMvdtS/1UyPB0dHcKIfrlczrVr16irq0OpVPL48WO++eYbYmJi\ncHd352c/+xnh4eE8ffpUmL8zfvx4oVvpXWBpaYmLiwuFhYV9bmdmZsawYcM0NDdqbGxseP78udbf\nWbt2LTdu3ODp06dIJBIcHBywtLTku+++w9zcXDCsfJuTikUiER4eHqxYsYKf/OQnVFdXU1BQQE9P\nD62trb1ErlKplJEjR2JmZkZ1dTUnT56ktbV1QO9pYmKCjY2N1uszEDo6Orhw4QKnTp1i0qRJbN68\nWWeHXl5eHlZWVri5uTF9+nTCwsIICwvD29ubJ0+esHz5cqqqqkhJSdEIeODFNQoKCmL79u00NTVR\nWFjIjh07+PbbbwX/suHDh7N48WI+//xzAgMD+fjjjykvL+fQoUM0NjZqHEtraysXL15k2bJlWj/3\nL2d4rK2tGTZsGKWlpTqvg65yVn19PVZWVlo1hDY2Nnz44Ye0trZqDCnURlJSEs7Ozr38/16loqJC\nKHeJRCIWLlyIpaUlp0+ffmvBrYE3wxDwGHhjOjs7OX36NAsXLqSxsZGuri4CAwP1/v1/lYBHm15E\nLUROTk4mLCyMTz75hLCwMIyNjXn27BkikUhoR7ewsMDX11fofnsX6FNigP51PNqwtbVl2bJlnD59\nmurqau7du8e6deuQyWS0tLTg6+srOFG/bXp6erh8+TJDhw5l0aJFODs7U1RUxBdffEFcXBx1dXXA\n/5a7Zs+ezccff4yjoyNfffUVDx48GFAg9ibt6UqlkoyMDPbs2YORkRG7du0iODi4z1JndnY2bW1t\n9PT04OzsTFZWFhMmTMDLy4uSkhJMTEz44IMPuHv3LoWFhb06tDo6Ojh+/DjOzs78/Oc/x9zcnIKC\nAn7zm99QV1dHTU0NY8aMEbJyXV1drF+/Hn9/f/bv309mZqZw/c6dO8fEiROFlv9X6e7u1iiF99Wt\nJZPJePLkiVaDTrV+RxcmJiasWbMGV1dX9u/fLwi6X93HgwcPmDdvns79wIsgrqenR2PopVgsZunS\npYhEIs6dOzco962BgWEIeAy8ESqVirNnz+Lv78+oUaNISEhg5syZAxLU/l/v0mpra6O5uZn09HSU\nSqXGtamoqGDhwoV89NFHjB49WuO1oqIifHx8NBay0NBQ0tLS3tmX56hRoygpKelXm/A6AQ+88ESa\nOnUqR48exdfXF1NTU7q6ujA3N+fixYv09PS8dS8qmUzGsWPHMDU1ZdWqVTx9+pSgoCDWrFnDxx9/\njJmZGceOHePgwYNcv36dzs5OgoKCkEgkzJw5kw0bNnDv3j2+/fZbmpub9XrP1w14KisrOXDgAI8e\nPWLDhg3Mnz+/389Oc3MzdXV1ODk5UVdXR3t7Ow4ODtjY2Ahu8z09PVhZWbF69WqePXumcR82NTXx\nzTff4OHhwZIlS7C0tGTOnDlCxufvf/87N2/eFLaXy+UcOnQIpVLJ5MmT2bBhA8nJyZw9e5akpCQU\nCgXTpk3TebxyubxXwJOfn681S/L48WM8PT21XoPq6up+LSVEIhGzZs1i1qxZHD16lNzcXOE1pVLJ\nxYsXmT17dr8PTerszqtBp7pc29HRwcWLFwfdR81A3xgCHgNvxM2bN+nu7mb27Nnk5uYiEom0igf7\nYrC6tNRTlt9lhmf37t0MHz4cOzs7bty4wZAhQ3oFKy0tLVqfzouKijSmLsML/Ye5ufmgmTf2h5mZ\nGR4eHjx+/LjP7dTC5Ve/4LVpeF5l1KhRdHd309LSQk5ODubm5oSHhwuaiLcZ7LW3t3PkyBHs7e2J\niopCLBZTUlKCt7c38MIde+bMmfzsZz8jLCxMmBlz48YN4TwcHR356KOPcHNz4+uvvyY9Pb3fhU0d\n8Oi7AHZ0dHDp0iVOnjxJSEgIW7Zs0dsfSn0N3dzcsLa2Jisri/HjxwMvshxOTk5CcGpqaoq5uTnn\nz5+nubmZmpoavvnmG0JCQpg1a5bGfWptbU1UVBTr1q2joKBA+LnabuKrr74Srs/27duRy+XcvHmT\nsLCwPh+Ienp6NLRRVlZW2NnZUVxcrPXctJWzYGAeWoGBgaxfv55r166RmJiISqXi7t27GBsb9xJV\na0MtWNaGVCplzZo11NfX6+0dZ2BwMAQ8Bl6b4uJi0tPTWbFiBSKRiMTERMLDwwfcRTRYGR61/uKH\nmFOjjba2Nn75y1/S3d0ttOvb29szY8YMZs6cyZQpU5gwYYLWQWV9uaOHhoZy7969H+IUtKJPWWvI\nkCGYmpr2moisrTX9VZKTk5k4cSIqlYpbt27R0dFBQEAACxYswMXFBYVC8VZ0TC0tLRw6dAhvb2/B\ny6uhoQGlUtlrqrVEIhF+vm3bNlQqFQcOHODYsWPk5eUhEomYNm0amzZt4uHDhxw9erSXduVl1DYU\n/WWEVCoV9+/f58svv0QsFrNz505By6QvWVlZtLe3Y2pqip2dHbW1tRoPJV5eXkIwUVVVhYeHB5Mm\nTeLQoUMcOXKEiIiIPoeHOjk5aQ06mpqaBJ2SUqmkpqaG9957j7i4OBISEnQGrq9meEB7Waujo4Py\n8nKt5SyVStVvSUvbeXz00UeUlJRw7NgxkpOTtYqqtdGfQ7qxsTEffPABT58+JTExUe9jMvB2MQQ8\nBl6L5uZmzp07x/LlyxkyZAhZWVmYm5sLT8YDYbC6tN61aahCodBY2JVKJY6OjkyfPp1p06Yxe/Zs\nFixYoDWo6csdffTo0VRXV2sV//4Q+Pn5UV5e3u+0YG1lLRMTE4yNjXUKfRsaGsjJyWHatGnMnTuX\n9vZ27OzsMDU1RSQSMXfuXAAOHz7c7/DAvnj+/DkHDx5k3LhxGpkLtTv6q4ucetpzeHg4tra2GjYW\nqampgo2FqakpW7duxdvbm3379nH37l2twZ1IJBKMRHVRXV3NgQMHePDgAevWrWPBggUDzoQ+f/6c\nxsZGPD09qa2tpbu7mzFjxmgEFF5eXoIouKqqCicnJ6ysrGhra8PGxqbX9GJtvBzcGRsbY2xsjL29\nPWfOnEEul3PlyhU8PDyYM2cOH3/8MZWVlRw8eFBrtk8ul/e67wMCAigoKEAulws/e/z4Md7e3lo/\nI42NjZiYmPTqJOsPS0tLNmzYIJT19CnNK5VKqqur++0aNDMzY/369eTk5JCSkqJ1m7KyMmbOnMmI\nESPYvHnzDzKR+18JQ8BjYMDI5XKio6OZPHkyHh4eKBSK187uwOBleFpbW99pOWvo0KF8+umnGBsb\nY2pqSlhYGA8ePODcuXP9TnstLCzU6Y4ulUoZP378O2tRNzY2xsfHR0PvoA1dOp6+ylqJiYmEhYVh\nbm5OSUkJ5ubmNDQ0CPtRL0AjR47k8OHDrzXRtqamhsOHDzN16tReU5LVAc+rPHz4ECsrK43XXrWx\n6OjoYO/evURHR+Pg4MCWLVvIycnh0KFDWoNTXTqezs5OLl++zLfffsuECRPYunXrgDIVL5OTk4OJ\niQnBwcGUlZVRWVkplLPUODs709jYSHt7O1VVVTQ3N3Pt2jW2bNkiuNL3x9y5c/n8889ZtWoVdnZ2\nSCQS/Pz8aG5uZvfu3ZSVlQnC3yFDhrBu3TpGjx4t6JFeDgoVCkWvdn9LS0scHR0pLCxEpVKhVCrJ\nyckhICBA6/EMpJz1Knl5eZibmzNlyhQOHDjQZ4cYvJgWrc5o9oeFhQUbN27k3r17vYIelUrFjBkz\nSE5OprKyklOnTrFjx47XOgcD2jEEPAYGzNWrVxk6dCjvvfce8GL+jloA+ToMlobnXet3AP74xz/y\n//7f/+PEiRMkJiby6aefYm1tzddff821a9e0lmZUKpUgWNbFxIkTyczMfGNvoNdFnyGE7u7uWqfk\n6mpNr62tpaSkRHCFf/ToEXK5nKioKE6fPk1TU5MQUIeHhxMQEMDhw4cH1BZeXl7OsWPHiIiI6LXw\nK5VKnjx50ivgkcvlJCUl9dKwvIyDgwMLFizgs88+Y+TIkdy4cYPjx48zcuRIfHx8OHDgACkpKRpl\nHLVzuhqVSsXDhw/58ssvAdi1axfjxo17o0GTmZmZyGQynJ2d6ejowM7OTmu5zt3dneLiYioqKigp\nKWHr1q04OzuzcuVKSktLuXv3bp/vY2pqiqmpKT4+PtTX1+Pk5ISFhQVr166lubmZtrY2UlNThftV\nJBIxadIkNm7cyO3bt4mJiRE+CwqFolfWRqVS4erqyo0bN/jTn/7EiRMnqKys1Nkqro9gWRsv20e8\n9957LFu2jJiYGNLS0nSWYfsrZ72KlZUVY8eOJT4+nt///vecOXOG+/fvk5eXR1VVlSDOlslkXL9+\nfcDnYEA3hoDHwIB49OgRJSUlLF68GJFIRE9PDzdv3uxzAmlfyOVylErloLiZv+sMjxorKytCQkIQ\niUSYmJgwc+ZMduzYQVdXF7t37yYlJUUjVf9qO7o2hg4diru7u9Du+0Pj4+NDbW1tn07kNjY2KBSK\nXtksXZ1aCQkJTJkyBRMTE+rr62lubiYoKAg/Pz8mT57MiRMnNIa4zZgxg+DgYA4dOqSXI3pxcTEn\nT55k6dKlWjUnVVVVWFlZ9bpn0tPTcXJy0mtRe9nGYsWKFTQ0NJCSksKIESPIysrim2++EXRNTk5O\nPH/+nK6uLmpqajh48CDp6emsXbuWhQsXvvFDgLojy9/fn9raWoyMjHTaQnh4eHDjxg1EIhEffvih\noDEyNTVl7dq13Lp1S0OYrAsjIyN8fHywt7fnzp07JCQkMHbsWMzMzKirq2P37t1kZGQIgZ+DgwPb\ntm3D3Nycf/zjHzx9+rRXwKNUKtmzZw+pqak8f/6c9vZ2urq6GDlypM7BjwPV76i5du0agYGBQnlK\nnyGFfQmWdeHm5oZUKkUmk5GTk8PFixe5fPmyxvmIxeI+H3oMDBxDwGNAb2pra7l27RqrVq0S0rfp\n6ek4Ozu/9tRbdXZnMOwS3rWGR01LS0svJ/MhQ4awaNEiNm/eTFlZGbt37yYzM1Mju9PfNVGLl99F\n14dUKsXf379PN2uRSKS1rKUt4CkvL6e6upqQkBDgxdwYsVgsTOueNGkSLi4unD17FkA456lTpzJh\nwgQOHTrUZ5kwNzeXs2fPsnr1ap2LSHFxca/sTldXF7du3dLLKuFlXrWx8Pb2pqenh6amJvbt20dC\nQgIikQgHBwfOnTvHsWPHGDNmDB9++KFOl/KBkp2djUQiITg4mMLCQrq7u7Xqcbq7u8nLy6OtrQ1f\nX99eupdhw4axatUqYmNjtc6qeZVRo0ZRV1dHR0cHSqWSxYsX4+npibm5OatXryYrK4u9e/eSn5+P\nSqXCyMiIBQsWsGDBAs6cOUNXV1evhT8gIED4PIjFYtra2nR2Z8HrlbSKi4t58uRJr4c39ZBCdUff\nq9oxfTI87e3t5OXlcfXqVfbt28eJEyc0HnLs7e359NNPOXfuHJaWlpiYmDBixAh++tOfGoYWvkUM\nAY8BvZDJZERHRxMRESFMc+3u7ub27dsDXgxeZjBtJX4MJS11R5GFhYXW1+3s7FizZg1RUVHcu3eP\nr7/+mqysrF7t6Nrw8PBApVLpZa44GOhT1nJzc+t1fK9qeFQqFd9//z3Tp09HKpWiUql49OgRUqm0\n1+Tal/2u1EyePJnQ0FAOHz6stTPqwYMHXLlyhQ0bNugcdgdotKOruXPnDl5eXjonGOuDmZkZYWFh\n7Nq1ixUrVuDp6cmtW7f43e9+R3V1NS0tLezcuZMJEya8tcBfpVKRmZmJQqHA09OToqIivL29e2VE\n2tvbOXz4MLa2toJhpzZcXV2ZP38+J06c6LeEOHLkSMrLy5HL5YLYfN68eRQWFiKTydi0aRNz5szh\nxo0bHD58mMrKSgB8fX35+OOPUSgU3L9/X6PsGR4ezuTJk4Vuuba2Np2Ba1tbG3K5XGv3oy7UgycX\nLlyoVQRtYmLC6tWrcXd3Z9++fULgl5CQQGNjo8b9oVKpaGho4OHDh1y4cIHdu3fz97//nfv372Nu\nbi7onezt7RGLxYwaNYrW1lbq6uqYPXu2oFlLTU0lNzeXP/zhD9y7d08jQDLwerz9OoKB/3OoVCpi\nY2Px8vIiODhY+PmdO3fw9PR8o8XgX2HooIWFRb/dHu7u7nz44YdkZmZy/vx57ty5w9ChQ/t8ShWJ\nREKWx8PD4y0fef94enrS3NxMQ0ODzoXS3d2djIwMjZ+93JouEokoKSmhtbVVmHdSW1tLR0cHYWFh\nGgGARCJh1apV/OEPfyAzM5Nx48YJr02aNAmJRMKhQ4fYuHGjYMiamprK3bt32bx5s1aTVjXqstLL\nOrSOjg7u3r3LRx99NPCLowW1jYWZmRltbW00NDTQ09NDbW0tubm5BAcHa11sX4fq6mq6uroICgpC\noVDQ0tLCihUrNLZpbGzk2LFjBAYGMmPGDPLz8/tcVAMDA2loaODEiRNs3rxZ57GquxPHjRtHTk4O\n9fX12NrasnjxYmJjY9mxYwe+vr74+Pjw8OFDTp06hZubG+Hh4djY2CCRSHBzc+PAgQPMmTNHaMOf\nMWMGcrmc27dv4+7urrMMXl1djZOT04CCx8TERFxcXPp80BCJRMI8KPVgzIcPH2JiYkJtbS1lZWXC\nP3V2083NjdDQUCG4eZn3338fhULB2LFjKSoq4tSpU6xfvx5HR0fs7e1RqVRIJBK6urqIj4/n+++/\nZ+7cub20Zwb0x5DhMdAvKSkptLa2EhERIfyss7OTO3fuMGPGjDfa92AJluHHkeHRVs7ShVrj4+np\nyahRozh27Fi/HV3BwcGUlpbqpWF526hLDX1leRwcHGhtbdVor1V3rbW0tAhWDS9P587KykKlUmk1\nnzU3N0cikRAfH9+rwykkJITp06dz+PBhnj17RkJCAhkZGWzZsqXPYAde2H44OztrZEBu375NQECA\nzmBuoMhkMr777juOHDnCuHHj+Ld/+ze2b9+OSqUiPj6eP/3pTxo2Fm9CdnY2IpGI4OBg0tPTkUql\nuLq6Cq9XV1fzzTffMGnSJGbOnIlSqaSrq6vP2UHwooRob2/PuXPntJZSVSoVFy5cwM3NjebmZiZM\nmMCdO3eAF3oYPz8/rly5Ary4f8aPH89Pf/pT7O3t2b9/P9999x1KpZKAgAA2b97MnTt3OHPmjCBo\nnjx5Mvn5+fz2t7/Fy8uLmTNncv78eY2yz0DLWdXV1Tx69Khf+wg1vr6+jBkzhocPHwIvgmW14ayv\nry9bt27ls88+Y8WKFYSGhuLo6Kj1gScoKEgI8n18fFiwYAHffvst9fX1wIvvAzs7O+CF1lEul7/R\nKAYDhoDHQD88efKE1NRUVq5cqfFElZqaip+fX78LSX8Mto/Wu87wDDTLVFhYyMiRIwkNDeWTTz7p\nt6NL3XKcnp7+Ng9bb/ora4nFYkaMGKFTx5OXl4dKpRLai9XlrGHDhum8t8RiMQsXLiQ6OrrX4L7x\n48czc+ZM9u3bR25uLlu2bNGrtPGqfqe1tZX79+/3aYGgL+ry0p49e+ju7mbnzp1MnDgRsViMk5MT\nw4YN4/3330cqlVJTU8PRo0c5ePAgmZmZr1XGUL+fkZERLi4uPHjwQEMXVFxczLFjx1iwYIGgmXr2\n7BlDhw6lrKysz0nWIpGIyMhIOjs7tXYQ3b9/n8bGRqKioigtLSU4OJicnBzh3p09ezYVFRXk5eUJ\nv2NsbMy0adPYtWsXSqUSpVJJQUEBw4YNY9u2bVhaWvLVV1+RkpKCn58fMTEx5OTkUFpaSmJiIhs2\nbGDatGlCUD2QgEepVHLhwgVmz56ts+zc2dlJfn4+8fHxHDhwgD/84Q9CEKfG3d2dRYsWMWbMGIYN\nG/ZapcmAgADB4kL9kKPOOIpEIjw9PZk6deqA92vgfzEEPAZ00traSkxMDFFRURqLRnt7O+np6Uyf\nPv2N32OwNDxyuZzu7u5Byx7pS0tLi94Bz6vt6Pp0dMGLzMb9+/ffSY3f1dWVrq4uamtrdW6jTcej\nbk1PSEjQaPeurKxELpdrlKteRSQS4ePjw3vvvceJEyc0WvPVreVWVlZ0dnbq3bL+qn7n5s2bjBs3\nTu/snC7q6uo4fPgwqamprFq1isWLF/daWN3c3DAzM2PHjh1YWlpiamrKyJEjefToEX/5y1+Ij4/v\n147jZSoqKlAqlYwdO5bW1lYaGhoEM9/MzEzOnTvHqlWrNATMVVVVjBgxAisrq35d3KVSKatWreLx\n48ca5cpnz57x/fffs3z5ciwtLXFzc6O6uho/Pz9hO2NjY5YuXUpcXFyvbIWFhQULFiwAXkxpjmiP\ncgAAIABJREFU3r17N9nZ2URERLBw4UKWLVtGVVVVr1EMbW1t3L9/n23btgH/W9LShzt37mBmZqaR\nTWxqaiIzM5NLly7x5Zdf8sUXX3Dv3j2MjY0JDw/n5z//OeHh4Rrlz4cPH5KRkcHcuXOxtrYmODiY\ny5cv63UMLzN27FimTJnCkSNHaG1txcfHBwcHB7Zv305DQ8M768r8v4JBw2NAKwqFgjNnzjBx4sRe\nQs5bt24RGBgotK6+CYOV4VFrZwaj+2sgtLa26r1o6mpHV3d0TZo0iRs3bnDv3j3Cw8MJCgoStndw\ncBB0ID8kIpFIyPLo0nK5u7tz7do1jZ8NHz6cx48fY2FhoXF/qWfvBAUF9fmeSqWS9957j2fPngkL\nuPqeVSgUfPzxxxQUFHDs2DE++OCDPjufWlpaaG9vF7IC6mnPP/3pTwdyKTTo6uoiKSmJR48eMWPG\nDCZMmKBTx+Xq6srTp0+ZOHEiK1euJDc3lytXrhAcHExERASPHj3iwIEDODo6MnHiRPz8/PrUhGVl\nZaFQKAgMDOTBgwdIJBI8PDxISUnh7t27bNy4EXt7e43fUU9YNjMzo6SkpN+uS3Nzcz744AMOHjzI\nsGHDcHNzIyYmhvDwcKEMM2rUKPLy8nj//fc5ceIE7733HhKJBFdXV8aMGcOlS5dYtWqV1s/ookWL\naGtrIz4+ntTUVKytrWlra9OZfZLJZJw9e5aysjLa2tr0yjw3NjaSnJxMVFQU6enpgv5GoVAI+pvx\n48drLUm9//77jB49moMHD7Jx40ZhNIe6tJyVlcWqVas4e/ashhRAH0JDQ+nq6uLo0aNs3ryZn/zk\nJwCsWrWKI0eOCBofAwPHkOExoJXr168LqeaXaWlp4eHDh28ttTqYAc+71u/AwEpa/bWja+voUnsg\nqV3U3wWBgYHk5OTobI93cXHh2bNnGk/m1tbWPH36VGM6t1KpJDs7mxEjRugsL8CLkpZa8Lxw4UI6\nOjq4fv06x48fF4wajYyMGD16NAsXLuTbb7/VGPD3KiUlJXh6egqLWlJSEqGhoQO2JYAXWbrs7Gz2\n7NlDZ2cnO3fuJCQkpM8A5eWJyyKRiNGjR7Njxw5aW1s5deoUvr6+fPbZZ4wZM4Y7d+7wxRdfkJCQ\noNWHS30NraysGD58uJBZycjI4OHDh2zdulXrYlldXY2zszNeXl6UlJToda7Dhw9nxYoVxMTEcOHC\nBWxsbDQEtX5+fhQXF2Nra4uNjY1GGWvGjBl9ZizMzMxwdXVly5YtzJw5k/379/erXzEyMiI2Nlar\nQFiNXC6nrKyMW7dusX//fuRyOVevXqWqqgovLy82btzIL3/5S1atWsWkSZNwdnbWua/KykpcXFyw\ntbXl0aNHdHV1abze0dHBf/zHf/R5zLqYOnUqvr6+HDt2TNivg4MDc+fOJTo6utd7GdAPQ4bHQC9y\ncnJ4/Pgx27dv77X43rx5k/Hjx781bYxMJntjHZA2fgwdWjAw0XJRUVGfJo1q1B1deXl5xMXFYW1t\nzaxZs2htbaWqquqtzXHRF/UTcFVVldbMgFQqxdHRkYqKCkEnU1NTg1gs1mgTf/r0KSqVSudwPDUi\nkUgIrqRSKZGRkfzjH//A1dWVZcuWaSxQo0aNQiKRcOLECVavXq21Lf1lO4na2lqKi4v55JNPBnwd\nnj17RlxcHJ2dnaxYsaLPFviXsbW1pbOzUyNIt7CwYPny5Tx+/JiYmBhGjRrFrFmzCA4Opra2loyM\nDL766ivc3d2FLKxIJBJKh+PGjaOkpERo86+urmbLli1aS7xyuZy6ujocHR1RqVTExMT0cizXhYeH\nB8HBwdy9e5edO3dqfF9YWFjg5OREUVERkyZN4tatW4wePRqRSIRUKiUqKoqjR4/i4eEhlMzVZVl1\nB5hIJMLf359Ro0bx/fff93ksKpWKuro6jc+QTCajvLxcyN5UV1dja2uLmZkZRkZGbN++fUDt6y/z\n8sDB4uJirUGINusQfZk1axZdXV2cOHGCdevWYWRkxJgxYygvLyc2NpaVK1e+8wz2PxuGDI8BDerr\n64mLi2PlypW9vhwbGxvJzc1lypQpb+39fkhbiebmZj755BOWLl1KcnKyxmsqlYqSkpJ+9QsDRd/A\nqy93dG2IRCICAgLYuXMnfn5+HD9+HDMzM27duvWmhzxg1GWtrKwsndu8rONpbm4mPT2dnp4e4uPj\nuXDhgmCroFAoNJy8db2fOuBpbW3l9OnTBAYGUldXR1VVVa/tfX19iYqK4tSpU720ROq/u7qs9vK0\nZ33p7u4mPj6eQ4cO4e/vz/bt2/UOduDFZ+7UqVOEhIT0WtT9/f0FDdfevXspLS3VsLHw9fXl+++/\n529/+xu3bt3i/v379PT0EBgYSFpampBBXb9+vc7PWV1dHTY2NhgbG2NiYoKjo6Pes51aW1vJzs4m\nKCiI2NjYXjoydVnL19eXjo4OjUybo6MjYWFhwt8fEOYsvZpVCQ0N7fdvolQqsbKyQqVSERcXx1df\nfcWf//xnUlJSEIvFTJs2jV/+8pesW7eO2tpaVq1a9drBDmgOHJw7d26vrKREInmjLlaRSMSCBQsY\nOnQo0dHRQifavHnzaGlp0WlAakA3hoDHgEB3dzfR0dHMmjVLa5YgKSmJkJCQ10r16+KHNA5dvnw5\nX3/9NbGxscybN4+SkhLa2tpITk7mT3/6E8eOHes1M+ZNUKlUemd4+nJH7wuJRCJ0dHl6epKXl8fl\ny5e1dnQNJuqyli6Nhbu7u9CptW/fPrq7u1GpVKSkpFBUVIRCoSAvLw8fH59+MwvqgKexsZGDBw8S\nFBTEkiVLWLx4MdHR0Vpb9H18fFi+fDnR0dEaJZu6ujpMTEywtramoqKC6upqJk6cqNc5q1QqcnJy\n2LNnD21tbezYsYOwsDC9HLZf5pNPPiEzM5PHjx+zaNGiXqJcMzMzli5dyoIFCzh//jyXLl2iq6sL\nY2Njxo8fz7Zt21ixYgX19fVkZ2djZGREUVERhYWFiMViIiIi+rymr2YF9S1rqedzjRs3jqVLlzJ0\n6FBiY2M1SpujRo2isLAQpVJJaGhor+6m999/n66uLqHLUB3wvMqsWbP6PBaRSMTQoUPp7Ozk+fPn\nWFtbExkZyeeff86mTZuYOXMm3t7emJiYcO3aNYKCgt4oEyqXy6mtrRX2sWbNGoKCgoSgx9zcHBsb\nG37729++9nuoz2vJkiVIpVJiYmIEG56VK1eSmprKkydP3mj//2oYSloGgBdfXhcvXsTFxUVrh8yz\nZ88oLCx8rVR/XwxWl1ZbW1uv8kpaWpqwmEilUg4ePIhUKtXIGDQ0NJCVlYWFhQUWFhaYm5sLs18G\nikwmQyKR6BXE9OWOrg8mJibMnTuXlpYWqqqq2L17N1OmTCE0NHRQfMpexdbWFktLS8rKyrQOQXR1\ndaWqqorW1la6u7uRSCTI5XJEIhFBQUGCI7U+wYZIJOLZs2ecP3+eqVOnCq3Vfn5+PHv2jJMnT2od\njOfl5cXKlSs5ffo0UVFR+Pj4UFxcLGTVvv/+e6ZNm6ZXKae+vp4rV67Q1tbGsmXL9DLOVSqVpKam\nCnYKRkZGdHR0UFtbq2EY+Ze//AVHR0csLS1RKBR0dHQwY8YMRo4cyY4dO4iPj2fv3r1ERkYKmi8X\nFxc6OjrIz89nxIgRXLp0CbFYjEwmE0TEutAW8MTFxfV7PqmpqXR1dTFjxgxhYT5y5AiJiYnC9PUh\nQ4ZgZ2dHaWkp48aN4+bNmzQ1NQkND2KxmKVLl3Lw4EG8vb2RyWSIRCK6u7upqamhqqqK0tJSioqK\n+Oijjzhw4ECvoEgkEmFsbMzvfvc7nj59yvr163Xe80VFRZSVlb2xC3ltba2QFYMX+qFf/epXggGo\nn58fGzZseCuNHWKxmOXLl3Py5EkuXrzI4sWLGTp0KFFRUcTExLB9+/YfRfn+nwHJf/7nf/7nuz4I\nA++etLQ0iouLWb16tdbFPS4ujlGjRvXyGnpTbt++zYQJE956WSsjIwMPDw+Njqd79+5RWlqKQqHA\nyMhIq/BaPeCrqKiIzMxMUlNTSUhI4M6dOzx8+JCcnBwKCwspLS2lsrKS/8/eeQdHdaZZ/9fdauWc\ns5AIkggCISSyQCJnjAwGTJIBGQ+2d2ZqZ7dma3dqvqmZqd0t19TsgBcPtgGTsQQmmGQkIYKyhAAF\nlHNq5Sx1/v6g+i5tRbBkj21OFQVI3X1v971933Of5zznNDQ00NbWRnd3N0qlEq1WK5Co1tZWiouL\nCQkJGXJfdSX4xYsXDynWHQmsra15/PgxO3bs4MmTJ8THx2NiYoKTk9OY9/v7+vqoqKjA19e33+8M\nDAzIzc1FJpPh4uKCo6MjMpkMsVjMqlWrSE5Opq2tjbVr1w67n4mJieTl5bFixYp+5FyXQJ6Xl6eX\nv6SDtbU1Hh4eXLhwAbVaTVlZGVOnTqWzs5Ps7GzWrVs3ZIVGoVCQkJDAjRs3mDlzJuvXr8fGxmZE\nn49Wq+Xs2bMUFxdTUlJCQUEBZWVleHh4kJWVhVqt5g9/+ANisZimpiYaGhqEEFVnZ2ecnJwwMjLC\n19cXR0dHvv76a2QyGV5eXkilUhISEqivr6e3txepVEpwcDD19fVkZGTQ3NyMhYUFFhYW/T6ThIQE\ngoKChEqkhYUFcXFxBAYGDkrWa2truX79ul6rTCKRCMaCJiYmwtRbX18f5eXlTJkyha6uLurq6vQm\n83Q3FA8ePMDV1ZXc3FyKi4tJTEyktLSUpqYmtFotZmZmTJo0ib6+PlpbWzE2NkYsFrNp0yYOHTpE\nTU0NfX19eHp6DnhMFAoFZ8+eZc2aNcOSwOHw7Nkz4f3qKuPPnj1j+vTp/OpXv2L27NmjeiMnFovx\n8/MjLS2N+vp6JkyYgK2tLSqViocPHxIQEPDSlcWfI15XeF6Dqqoq7t27x969ewe8u62rq6OyspIN\nGzaM+ra/Tw3PuXPn+Ld/+zdycnL46KOPsLa25uTJk3ourbNmzernL6TVaunt7aW7u5vu7m56enqE\nfzc1NVFZWan3c12bzsDAAIVCQXR0NKampnpVI92/zczM6OzsHDYdfaRwdXXFwsKC1tZWtm7dSkVF\nBXfu3CElJYWlS5f2sxgYTUydOpWjR4+yevXqAUmzs7MzeXl5vP/++5iYmJCXl4dGo8HW1paSkpIR\nXbTLysro6elh7dq1A46u64zxvvjiC71Kw4vw9PQUCAI8n6aRy+WEh4cPWsnTarXk5+dz+/ZtPD09\nee+99176rlqtVmNkZCS083SIiIjAxcWF1tZWfv3rX6PRaDhy5AhdXV3Ce0pLSyM+Ph5vb28mTJjA\nxIkTOXDgAHFxcRw5coSVK1cKU1AhISE8e/YMc3NzpkyZQnh4OFlZWcTExGBiYkJQUBDTpk3D0NAQ\npVJJU1OTnqWAWCxm3LhxAhn8NhQKBRcvXmTVqlX9KhhmZmZs376dEydOYG1tjZeXF/7+/jx48EBo\na3366acsWrRIj0zNnj2bvLw8Hjx4gFarRaFQDNgefeutt/j444+JiYmhr6+PN954Q6+NlJOTw9df\nf42dnR1Lly7Ve193797Fw8NjVBLIdXq75uZmTp8+LbRRR7Pd/20YGhqybds2Tp48yd27dwkPD2fh\nwoXU1NRw586dETtF/5zxmvD8zNHd3U1MTAzr168f1EL/7t27LFiwYNRyfnTQarXI5fKXEoiOFAOJ\nhY2NjYmKiiIrK0vwq9GN1eraK+np6YhEIoKCgoQLqUgkElpbI7kz1Gg09Pb2kpGRQVVVFZMnTxYI\nki4n6kXypGt9HTlyZEBS9G2CpAtkHAzBwcGkp6fj5+c34ETXsmXLXjpJeiSwtrbG3t6e0tLSATOJ\ndLoqXSVh3rx53L17lxMnTqBSqYZtZ+Xn53Pt2jXMzMyGFHcbGBjw1ltv8dlnn+Hg4DDgou3l5SUk\nvctkMgwNDQd8HEBzczM3b96ko6ODjRs3vnRuWU9PD2lpaWRkZGBnZ6dnhmhnZ8fUqVMF7cv169eJ\niIggMjKSo0ePolKpCAoKoq2tjb6+PjQaDaWlpSQkJGBmZsaECROYN28eV69eRa1Ws2DBAhobG5k5\ncyZVVVX4+PhgZmbGggULmD9/PiUlJWRkZBAbG8u0adNwd3fH3t6+XwvI29ub0tLSAT+Tmzdv4uHh\nMejn5eDgwKZNm4iOjhZiPaytrSkvL8fHx4dx48aRlZXFxIkTqaioECaouru7USgUQqRCe3u7ngha\nKpXi5eVFZ2cneXl5GBoa6hEMmUyGn58fwcHBZGZmcurUKSZOnMjixYvp7u4mOzv7O7eydKiurmbB\nggWcOHGC7u5ugcCOtfmnToB+4sQJDA0NWbBgARs3buTTTz/F3d190GPyGs/xmvD8jKHRaLh48SIB\nAQEDtiHgefVHN9Ew2tAJL0e7FKvRaOjp6Rm0PfQiWfDz8yM0NJT4+Hg2bNiAs7MzKSkpHD58GD8/\nP+bMmfPS4ahisVjYtouLC1OmTBny8V988QXTp0/HxcWlXxWprq6uH0FSKBQCCfr23zpCVFtbS1FR\nER4eHhgZGTF58mTB8fb06dOMHz+esLCwUdEYvIgpU6aQk5PTj/C0tLQIE1RyuZx33nmHS5cuIZVK\nUSqViEQigoODB3XIffLkCbGxsbz99tvExMQMGX8AYG5uztatWzl16hQ2Njb99FxeXl7k5eUJC5Uu\noDI/Px9LS0tcXV1RKpU8ePCAjIwMFixYwOzZs19Ky9Xc3ExKSgo5OTlMnjyZ3bt3U1BQIDghS6VS\nli1bRnNzs6Ajy8/Pp7y8nHHjxrF9+3auXbvG0qVLkUqltLa2kpmZSVZWFs7OzoLmRaelgedtW41G\nw9q1a0lOTtbz0dI5VE+YMIG0tDRu3bpFRkYGEolEaKnqCISPj4+gN3rx+5Kbm0tlZSVRUVFDvnfd\n+XX27Fn27duHr68vGRkZNDQ00NPTw+3bt4UQUE9PT+bMmYOjoyPnzp2jqKgItVrNBx98wNGjRwX/\nHbVajZOTkzDVpVKpKCsr07M6mDJliiDknz59OomJifz9739HLBaPSssYnhPYnp4eHBwc2LVrF1ev\nXhWmz74Pt3NTU1N27tzJ8ePHMTIyIjg4mC1btnDq1CmcnJy+c7vup4zXhOdnjLt37wIMWPZ/8TGL\nFi0aE+HrWAmWe3p6MDY2HnBxGsgcb8GCBdjY2AieLRs2bGDp0qVkZmZy5swZ7OzsmD17NpMmTXop\nctbZ2TksWZLL5dTW1rJt27YRV9B0QlYdCXqRINXU1NDT04NUKuXixYtotVrUarUeMRo3bhzNzc18\n/PHHeHp6Cvk/OsJkaGj4ynqfKVOmcPfu3X4+LgkJCcyZM4cnT57w7rvv8tVXX9HX16cnQN28eTOZ\nmZn9xtJTU1NJSkpi165dODg4CMaDw8HZ2Zm1a9dy4cIF9u3bpzct5+rqKoiHra2tyc3Nxdramvj4\neExNTVm+fDl37tzBw8ODAwcOvFTERFVVFcnJyVRUVBAUFMTBgwcRi8VcvnyZ3t5e3n//fR48eEBV\nVRWTJk0iMTFRr0Jw6dIlPvzwQzw9PTl48KDwujY2NixdupTFixeTl5dHWloajY2NgmbM1dWVnp4e\nWltb+etf/4pSqaS3txeNRtPvvPX09BSE4yqVivT0dJ48eUJQUBBBQUHY29uj0WhoaWkRfLLa2tq4\nceMGb7/99rBVWaVSiZ2dHWZmZvztb39Do9GgUqkwMTEhMDCQnp4elixZonejVVZWRnl5OSKRCGdn\nZywtLQkICCAlJQWxWIyhoSENDQ0UFRUBz78HiYmJ+Pj4oNFokMlkepVLIyMjwsPDhXy2hIQEVCoV\nwcHBrzSEoEN1dTWurq56behVq1bR0NAwph5Y9fX1PHv2DH9/f5ycnNi5cycnTpwgOTmZ27dvs2LF\nCi5cuMD+/fvHpGr+U8BrwvMzRUFBAU+fPiUqKmrQRbysrIy2trYBU6tHA2M5kj6UvuLbi7nOR+ZF\nmJmZERoayvz588nLy+Phw4d88803BAcHExgYOKL91mXhDIVXGUeXSCSCAHUwdHR0cOTIEX71q18h\nkUj6aY+6u7txdXWltLSUK1euYG5ujlgspru7G41G06+d9u222ov/l0qlwmdqbm6Oq6srRUVFQiCo\nTCajtLSUNWvWUFdXx/nz5wc0aZPL5Xz00Ud89tlnwHNy+uDBAx4/fkxkZKRQjXpxqm44+Pv7Cz43\ne/bswcDAgKqqKjo7O5FIJJiYmLB69WrMzMz4+9//jlqtprOzkxs3bvDmm2+O2BdJF3iZlJREZ2cn\nc+fOZePGjRgaGlJdXU1MTIwQDimRSFi3bp3QRtVpb3RErq+vD5lMNmi8g4GBAZMnT6agoAC1Wo2h\noSGVlZVIpVLUajU7duzg/v37VFVVCSaCOt3P+PHjMTU1xcnJSfjeSyQSJkyYwLJly3j06BHHjh3D\n2dkZW1tbiouLsbOzQ6PRcOnSJebNmzfgot7T0yO0piorK2loaMDJyQkPDw80Gg3W1tY0NDQwffp0\nPD09EYlEpKam4uvri1arJTExkdTUVCZMmEBlZSWPHj3CxsaG9PR0IV9KJpORmJiIRqNBJBIhkUgE\nTVd3d7eQQ/YiWlpayMzMZP/+/SgUCr1oFp0J4sviRf+diooKenp6hDDYsURjYyMPHz4kOTkZqVTK\nhAkTmDVrFmFhYajVamJjYzlz5gxXr17lzTfffG1KOABeE56fIVpbW7l69Spbt24dtMSr1WqJj49n\n8eLF3+luaCh8n4JlHUa6UOogkUiYNm0a06ZNo7q6mpSUFO7fv09AQAAhISFDukSPxIPnu46jDwZL\nS0t8fHx48uQJISEhWFlZDWqy1tjYSFxcHPX19axduxY/Pz+hbP/tKlJTU1O/nwN6REipVBIXF0dr\naytmZmakp6czbdo0ent7UavVgy4MarVa8GPRarV88803lJaWEhkZqUfuXobwAIKu5fe//z0xMTFU\nVVUhkUjQarUsXbqUDz74QM9DSKctG0zT9iKUSiVPnjwhOTkZExMT5s2bh5+fn0BeUlJSePDgAevW\nrdOrXIlEIqECJhaLCQ8Pp6WlBSsrq2FDefv6+rhw4QKmpqbs3buXs2fPIpVKsbe3p7Kykjt37mBm\nZkZAQIDgLOzo6Ehubi7Xr1/H0dFRCKWsqqrCwsKCTZs2YWhoyLJlywgLCyMvL4/79+8TGxsriLql\nUinz5s1Dq9XS3t5OZWUlFRUVVFVV0dHRgbu7O56enixduhQ3Nzfh/SkUCk6cOIGZmRl5eXl4enoy\nZcoU7ty5Q2VlJUlJSXR3d7N//37u37+PgYEBoaGhgl4rKCgIOzs77OzsmDhxIp2dnXzxxReEhoZi\nbGyMkZERJSUl/XRpOk3U/Pnzhamt7du3U1ZWJmR0LVu27KU1WdXV1cLkZWJiIvPmzRszsqNQKGhu\nbqapqUmwblAqlSiVSp4+fYqdnR2mpqZ0d3djZGTEmjVrOHXqFCkpKcydO3dM9unHjNeE52cGpVLJ\nl19+SWhoKB4eHoM+rqioCLlcPqYiuH+UCs9I4e7uzptvvklHRwfp6ekcO3YMNzc35syZg7e3d7/X\nHS4pXZeOPmfOnFfan+EQHBzM9evXCQ4OHvI96zK6dBNdycnJLzXRpVAo9CpIbW1tfPPNN3R0dFBW\nVoZMJkOj0ZCXl0d9fX0/Y70XYWlpSWFhIZmZmXR0dLBjx45+n+HLEh5d2OhHH33Ub9vffPMN27Zt\nY/r06XqvKZFIaGhoGJQkdnd3k56eTkZGBu7u7qxfv16oXMDzc/vq1au0tbWxb9++QUfXtVotTU1N\nvPXWW+Tm5tLc3Dzke+no6ODMmTN4eXmxcuVKFAoFlZWVTJ8+HblcztKlS3F0dOTixYtUV1cL4nzd\n4h4REUFlZSVFRUU0NTUBz3VmJSUl+Pj4YGRkhIGBAQEBAfj4+HDo0CHq6+spLCzE3t6eL774gtbW\nVtRqtaC/mTVrll7F6NvQTRcdPXoUmUzGihUrkEgk+Pv7c+bMGWbMmMHmzZuRSCSCR1NxcTEWFhbI\n5XK9ipJUKsXW1haFQoG/v7+gOaqrq+tHeJ48eUJPT0+/hd/b25v9+/eTk5MjZG8tXbp0RNoXrVYr\nJMvX19cjk8l46623hn3ecK+pm/hsbGykqalJIDnd3d3Y2tri4OCAtbW1XqRKWFgYc+fOZdmyZZw6\ndQqFQkFjYyNbtmzhs88+w9XVdUT+UD8nvCY8PyPo/F7s7e2H9IbRarXcvXuXsLCwMS3TjqXp4FCV\nq+8KS0tLlixZQmhoKE+fPuXWrVvA89HagIAApFIpKpWKvr6+IUWSg6Wjjxa8vLwQi8V6ws7hHv/i\nRJdOMzLcRJehoSGGhoZ6i3pJSQm2trbk5OQId6FisRhHR0e8vLyoqKjQswOA5xMoixcv5urVqyiV\nSoyMjPjrX/+KgYGBXjuto6ODlJQUHB0dB2yzfbsiqdFoOHjw4IBEq7e3lxs3bhAVFcWcOXMwNTWl\ns7OTkydP0tjY2E983dzcTHJyMrm5uUyePJk9e/b0O351dXVER0czYcIENm3aNKT+raOjAwMDA2F6\nbSjn3MbGRs6cOcOsWbOYP3++0A4Ti8VMnjyZ6OhoVq5ciZGRESqViv3795Odnc2jR4+wtrbm3r17\n5OTksGbNGkpLS0lJScHZ2ZnFixeTmZnJ5cuXhQBRKysr2tvb0Wq1FBYWYm5ujomJCW1tbWi1WmbP\nnk1gYOCIA3otLCzYvn07n376KY8fP0YkEpGTk4NWqyU0NJTm5mbS0tKoqamhq6sLpVKJl5cXhYWF\nVFRU6FVhNBoNcrlc79pRX1+vR2y6u7uJjY1l+/btA17DdIaX/v7+pKenC7EgixcvHvImpbm5GWNj\nY8zMzLh58yZz5swZsb5Ro9HQ2tqqR2p0/xaLxTg4OGBnZ4e9vT0TJkzA3t4eKysrvf3oLYyLAAAg\nAElEQVR//PgxBgYGbNu2TdAH+vr68sc//pGSkhK+/PJL3n77bTZu3MjFixfZv3//a1PCF/Ca8PyM\nkJWVRU1NDfv27Rvyjj8vL08I7RtLjGVS+lCtptHqbUulUoKCgpg5cyZlZWWkpqYSHx9PYGAgvr6+\ngi5mMAyXjv5doZt8Sk9PH7FhpC6j67tOdE2dOpWkpCShNdbQ0AA8nzD553/+Z/7f//t/tLe309fX\nh1gsRiKRsHHjRqytrXF0dGTr1q1C8KVcLu83uWZgYEB7ezu1tbX9ptikUqkeEaqsrNQbBf82+vr6\nuHLlCsuWLQOej9fv2bOHL774ArVazcKFC6mqqiIpKYnKykpmzZrFwYMH+y32Wq2WzMxM7t69y6pV\nq0ZUHa2rqxMm0ywsLAaMxYDnQugLFy6wdOlSZsyYIfz80aNHGBoa0tjYyKRJkzA1NaW2thYbGxsc\nHBwIDw9n0aJFPHv2jLS0NKqqqlizZg1ZWVnC9y8vL4+jR4/i6OhIUVGR4E8klUrRarXY2Njw3nvv\nCf+vra0lIyODw4cPCzoSLy+vYc9jFxcXfH19+frrr7G0tGTPnj2kpKSQkZGBj4+PXqyLQqGgtLSU\n8PBwrly5woEDBwQhbl9fH0ZGRsJ3S6vVUl9fr0fMb9++TUBAwLAiYgMDA+bOncuMGTN4+PAhR44c\nITg4mHnz5g0o/K2ursbd3Z2WlhZKS0tZt25dv8fI5XKampr6/WltbcXS0hJ7e3vs7Ozw8PAgMDAQ\ne3v7Efv37Nq1CxsbmwH90saPH8/atWs5e/Ysu3btIigoiJiYGHbt2jVmsoQfG14Tnp8JamtriYuL\nIzIyckiBrEajISEhgRUrVnwvzrxjRXgGK+WORoXn2xCJRPj4+ODj40NLSwupqamcPn0aiURCVVUV\n7u7uA36WI01H/y4ICAggPj5ez85/JHhxtDcpKYmjR48yY8YMFi5cOCLd1aRJk7h06RI2Nja0trYC\nzxeXyMhISkpK+Pjjjzl//jydnZ34+/vj6+tLS0sLnZ2ddHV1cfToUQIDAwkICBBG7XUk9v79+wQG\nBg64mN29e5cnT57g6OiIg4MD5ubmQjL7YNBoNDQ2Nur9zNLSkl27dvH555+Tnp6ORCJh7ty5vPHG\nGwN+f+RyOV9//TUNDQ288847QxLuF1FfXy/cqVtaWg5IeAoKCrh69SobN27UqzjpyN/s2bN59OgR\nq1evBhDOOR0kEglTp05l6tSpyGQyfv/73wvVrr6+PhITE7l58yYBAQGsWrUKNzc3DA0NuXfvHqmp\nqfT09PDRRx/h6enJxIkTmThxIhs2bGDFihU8efKE69evAxAUFMT06dMHPT/a2tqENppEIsHS0pI5\nc+Zw6tQp5s+fj729vfB7MzMzjIyMCAkJQSaTcfv2bdavXw88r8q9SBA6OjoQi8VCJaOoqIiqqqqX\n8twxMTFh2bJlhISEcPfuXQ4fPkxoaCgzZ87UIwu6hPSHDx8ybdo0ampq+hGbvr4+oVJjb2/P1KlT\nsbe3x9bWdkSxJUPB0dFxyN/7+fmhUCg4ffo0u3fvprq6mtjYWFasWPGdtvtTwWvC8zNAb28v0dHR\nrF69etj2ydOnTzE1NR1TR14ddBeG0cZYaXhGAltbW1atWoWLiwspKSl89dVXmJiYMHv2bMEjBF4+\nHf1VYWhoSEBAABkZGSxduvSln29kZERYWBizZs0iISFhRBldcrmcq1evAs+ntnp6elCpVKxcuRJ7\ne3uUSiXJyclMnTqV3/72tygUCk6dOoVIJBL0OY2Njdy5c4eGhoZ+Dt9DaXh0ztXt7e0UFhYKYYvf\nbp+9CBMTE70Wr1Kp5PHjx6SkpGBmZkZPTw9TpkwZVAslk8mIjo7G09OTffv2vdSiVl9fL+hsdJ+V\nWq0WzpNHjx5x9+5dtm/f3m9qS1eJdXFxIT8/XyD51dXV+Pj4UFtby5dffomxsTEWFhaYm5ujVCoF\nB3AdxGIxb7/9tt61oampibS0NHbs2MHJkyd5//33qayspLi4mPv372NiYiJMfr377rvU1NSQkZFB\nQkIC/v7+zJo1Sxjdhufk/vLly8ydO5f09HScnZ2Jjo5m+/btODo6kpOTIxAlkUiEh4cHDg4OiEQi\nVqxYwSeffEJhYaGQuv4iqXqxSqZQKLh+/Trr1q17JaNUKysrNm7cSH19veBQHhwcjKWlJc3NzeTm\n5mJqakpLSwumpqbIZDKB2EyaNEloQ/2QE1IBAQHCd2rbtm2cO3cOd3f3Yf3Afg54TXh+4tBqtXz1\n1Vf4+fkNe8Kr1Wru3bvHxo0bv5cv7I9hSutV0dvbi5eXFytWrKCoqIjU1FTu3LlDcHAwQUFBwl34\naLtXD4Tg4GCOHTvG4sWLX9lPycLCgnXr1jFnzhy90d5p06YJ54rOzyUxMRGlUklwcDANDQ2sXLmS\n4uJiZs6cCYCTkxOdnZ1YWVnR09PDqVOnCAgIwMnJiUuXLqFUKoHnraVVq1b12xedCFkHuVxORUUF\n5eXlFBcX6/1O156Kj4/n/v37AxrDiUQiIiMj+wmRN2zYgIeHh7CPGo2GZcuW6X03srKyiI2NZfny\n5a9k31BfX8/y5cuB/zOs7OrqwtLSkvv37/P48WP27Nkz4I1BZmYmFhYWlJSUMHPmTGG/qqurWbRo\nkaBHam9vRyaTCc/buXMnJ0+eRKFQYGBgwO7du/XIjkql4uLFi4SFheHm5oazszMymYwpU6YwZcoU\ntFotdXV1FBUVER8fT1NTE97e3kycOJH58+dTXFxMTEwMxsbGBAUF0dHRwePHj9m8eTMqlQoHBwds\nbW2pq6vj5s2bzJ49m+vXr2NkZCRMr9XW1rJ48WLgOenesGEDly5d4sCBA/T29updN15sZ8XHx+Pl\n5fXSN2x9fX39KjXt7e20tbURGxuLVCpl/PjxKJVKXFxcGDdu3IDtrH8UzJo1C7lcTnR0NGvXruXS\npUs4OTmNmV7wx4LXhOcnjvv37wvTG8MhKysLW1vb703ZPxYtLa1WS1dX1w9W4dFBV2USi8X4+vri\n6+uLTCYjNTWVw4cPC0GI3wfs7OxwdXUlJydHT//xKtBNdOnGn5OTkwkPD6ezs5N79+7h7OzM1KlT\nqa+vZ9myZfzlL3/Bx8dHb7u6Co2dnR3Hjx9n7ty5wqSanZ0d9fX1ODg40NXVRUVFxYAxFZWVlYIr\ncVNTE25ubnh5ebF69WrOnTuHRqPBy8uLt956C6lUytmzZ5kzZw6NjY3CKL2hoSEGBgYcPXqUpKSk\nQYXIZmZm7N69m1OnTnHr1i1WrlyJSqXixo0bVFdXs3v37mFbDQNBFyvyotjb0tKS9vZ2Hjx4QE1N\nDXv37h2QvHd2dtLU1MSCBQtITU3l/fffB56TfV3ieHJyst5zJBIJu3btwtnZGRcXFyoqKpDL5fzP\n//yP3uPi4uKwsrISoj58fHwoLS0V7BN0Joeurq4sWrSI7u5uSkpKKCoqIjY2FisrK6ZOnYqBgQF3\n796lt7eXadOmIZVKOXPmDEqlksbGRt577z2OHTuGVCqls7MTZ2dnMjMzsbGxwdXVVe8zHTduHJMn\nT+bmzZtMmDBBr6VVX18vtJdycnL4xS9+MeDnrdVq6ejoGFBfI5fLhUqNvb09AQEBQhtKLBaTnZ3N\nnTt3MDAwoLi4mAMHDrzMof5BMH/+fORyObGxsYSGhgqmhN/HTdY/Kl4Tnp8wdLk5UVFRw4rWlEol\n9+/f/84jli+DsZjS0uVSDdZW+L4qPLoL+ItwcnJi/fr1LFmyhEOHDvH06VNBg/GyLs4vi+DgYBIS\nEpg+ffqoED5PT08iIyOJjY3l/PnzGBoasmLFCmxtbTl+/DiLFy9GKpXi5+dHbm6u3gRNXV0dWq2W\nyspK1qxZo0eGdLlAb731Fr29vZw7d461a9diYGBAeXk5FRUV1NbWolar8fPzY/ny5bi5uelVrqZM\nmSLsj+69Ojo68p//+Z8UFRVx8+ZNysrKWLduHZ6enhQXF+Pp6UlUVNSg4+MmJibs2rWL06dPc/Hi\nRRobG3F2dv5OC4hMJuuXYm9ubs7t27cxMjJiz549gzrmZmdno9VqkUgkQl6WQqHgwYMHAHz++ee4\nu7tjYmJCd3c3UqmUd999FwsLC8HYcc+ePTx69IiCggKhOlVcXExeXh7vvvuusF8+Pj6CTmcg6Dx/\nAgIC0Gg0VFdXk5WVRXZ2NiKRiIkTJyKXy/niiy+E6l17ezslJSVERETw5z//mZiYGEG/oyPEarWa\n3/zmN8J2lixZwtGjRxGLxf1aWkuWLOHixYssX75ccGTWjXnrRrybmpowMjLSIzZ+fn44ODgMmCL/\nIqZPn057ezuPHj2iq6uLxMREFi1aNOIptR8KYWFhyOVycnNzcXFx4erVq0RERPxsTQklv//973//\nQ+/Ea4w+2tvbOXPmDG+++eaIsqB0GTzz5s37HvbuOZKTk5kxY8ao5Nvo0NbWRlFR0aBj9zKZjNbW\nVvz9/UdtmwNB5yI70ALa1tbGs2fP+PDDD5FKpSQnJ/Pw4UO0Wi0ODg5jEuNhY2NDUlIS7u7uLxWT\nMBC0Wi2lpaVcvHiR1tZW1q5di7OzMwkJCWRlZaHRaKitrRWObWpqKkFBQcLzL126RFtbG1ZWVv30\nOebm5igUCjIyMlAoFHR2dpKeni44986cOZOWlhZhWsna2rofUfT19dWbfuvu7ubkyZOUlZURGhrK\ntm3bMDExwcPDg6CgIGGKKysrC7FYjIuLy4DkU3dckpKSsLGxESbJXhXPnj1DKpUKFaze3l7i4+OR\nSqXs2bNnSC3QlStXMDMzo7m5GV9fX7Kysrh27RodHR3Y2dkhlUpRKBSsWrWKnp4eNm3ahJWVFSdO\nnKCxsRGJRML48eOF1k1AQADd3d2cOXOGN954Q6+6YmFhQVxcHIGBgSMid7obrQ0bNrBs2TLEYjEt\nLS16bTV4rkGqra3lv//7v2lubkalUqFWq4W/ExMTsba2Fr7LEokENzc3YmNjcXNzw9XVlfLych4/\nfkxdXR1tbW1UV1cTFxdHaWkpXV1dGBkZ4ebmxowZMwQriRkzZjBp0iTc3d2xtbUVWmnDITk5maam\nJnbs2EFLSwtff/01KpUKV1fXf9gpKF1+ms5dvL29HbVarSdq/znhdYXnJwiVSkV0dDRz584dkYuo\nQqEgMTGRnTt3jv3OvYCxaGnpErkHw/dZ4RmMWOjG0Q0MDPRcnFNTU0fs4vyyEIvFzJo1i/T09O90\nsdMtKB0dHYSHhzN58mRhsTA2Nuby5cvA83PqwoULREZG0t7eTktLC7a2thQUFFBeXs748eOprKwU\nxLNVVVWUlZVRUVEhGBVaWVmxevVqIeTWy8uLcePGjThLC55b/1+4cIHe3l4A0tPTsbGxwdjYmN27\nd2NhYSHEFcjlcu7cuYNCoRD0IzqoVCpu3bpFWVkZe/bsIS4ujqtXr7J+/fpXrszJZDLh+6m7QbG1\ntR12AW1tbaW1tRVPT08qKytJTU0lMDCQTZs2cenSJVQqFStWrBCOzYQJE1CpVBw9epTm5ma0Wq2w\nzwqFgqqqKrq6urhy5QozZszoJ6QXi8WMGzeOsrKyIUftlUolN27coKamhsjISKEtOHPmTGbOnMkf\n/vCHfsftypUrqFSqAY9nT08P//Ef/8HmzZtpb28XqjQAKSkpPHr0CHNzc0Hzs2LFCsaNG4eNjc2Y\nEJCKigpcXV3x9PTE09OTkJAQ4uPjOXz4MIsWLSIwMHDM4yVeBSKRiHXr1nHx4kVEIhEPHjwQ3sfP\nDf94R+c1vjNu376NhYXFiKs1KSkpeHt7v3Qq+HfFWBCe4fQ7MPYaHq1WO+SkWHFxcT9diru7OxER\nEbz33nsYGhpy7Ngxzp49S0lJyaiRtMDAQAoLC+nq6nrp5zY0NHDhwgWio6OZOnUqv/jFL/plEcXG\nxiISiTA0NEQsFlNTU8O1a9c4f/48rq6uTJ8+ndOnTyMWi/Hx8cHExIRPP/2Ujz76iAcPHiCRSAgP\nD+c3v/kNERER1NbW4uHhgY+PD1u2bCEmJkYIlxzJZ9LX18cXX3whkB2RSMT06dPZu3cvlpaWyOVy\nIZdMB93o/YtoaWnh888/p7e3l6ioKDw8PNi+fTsdHR189dVXwya3DwadO3BDQwPHjh1jxowZBAUF\nDXp8dHldZ8+eRaPR0N7ezowZM9i+fTv19fVcvXoVhULBgQMH+h0bXfVQNwmnVCrp6+sT8t5u3LhB\nb29vP6Kng07HMxhaWlo4duwYKpWKffv29RPHqlQqIajV399f0GzpsrAGQ09PD3/6059ISUmhra1N\n0B9ZWFiwfPlyZsyYgVQqFSYJ7e3tx4TstLa2IpfL9T4fGxsbIiIi2Lp1Kzk5ORw5coSCgoLv7abq\nZSAWi9m0aZNg/hkdHf1K14EfO15XeH5iePr0KaWlpezfv39EC3tvby8pKSns3bv3e9i7/4Purm60\n2zdDTWjB91Ph6e3tRSKRDOrXMtQ4+osuztnZ2dy+fRvQd3F+VZiYmODv78+jR48IDQ0d0XPa2tpI\nSEigqKiI+fPns2nTpgH3QTd6vn79eiFPq7W1lbNnz5KdnY1CoSAnJweATZs2kZ+fj5WVFdbW1kRF\nRfV7TT8/PzIzM0lNTWXevHl4eXnx5ptvEh0djZWV1bDHsbm5maSkJOF1lUolIpEIW1tbRCIRJiYm\nAhFycnKir68PPz8/Wlpa9L43eXl5XL9+nUWLFumNpeviEi5cuMDFixfZtGnTSy20us+nt7eXS5cu\nsXz5cgICAigvL+/nxdPc3ExWVhZPnjzB2tqa7u5uXFxcaGlpQaVScfz4cebMmUNwcDA3b94c0MRO\nKpWyefNmLl++LBByU1NTLC0tsbW1JTExkYMHDw76Hnx8fEhKSkKr1fa7ruh8gkJDQwkJCRnwumNg\nYMBvf/tb4HlV6dNPP8XT01M4BoPByMiIiIgIvfP12bNnhIaGEhcXJ2hvxiqeRYeUlBSMjY0HrIq4\nurqya9cuiouL9TK6Bgt//aEgkUjYsmULZ86cwcLCQjAl/EesSo0Vfj7v9GcAnUHXli1bRlw5SUpK\nwtfXd0z8cIaCrroz2tWW4VpaMPYVnqHaWSNNR5dKpcycOZP33nuPlStXUlhYyF//+ldiY2Npb29/\n5X0LCQkhMzNz2KpEV1cXN2/e5OjRo1hZWfHBBx8wb968QQlXYmIiU6ZMEfQRU6ZMYcGCBbS1tQlt\nK41GQ0tLC+7u7rzzzjssXLiQzs7OAV9TJBKxatUqHj58KLgke3t7ExERgUwm66cHAQQh9Pnz5zl2\n7Bjm5ua8++67GBgYsHnzZiZOnCic5y8SnjVr1vDhhx+yYcMGDAwMuHfvHmq1mlu3bnHnzh22b98+\n4EIulUrZunWr0EIeaOR9MDQ0NGBubs7Fixd54403BC8enfmgQqEgKyuL48ePc/z4cTQaDbt27WLD\nhg3I5XK0Wi1KpRIzMzMOHjzIwoULkclkQ7YrVSoVhYWFLFmyhF27djFnzhysrKyEuIKhCJsuMb2l\npUX4mUajIS4ujhs3brB161Zmz549ou/WjRs3cHNzY+HChfj4+AxJ4hUKhfDZ6NDb24urqytBQUHI\nZDKWLl06pou2VqslJydnwGlBHXTi7AMHDjB9+nQuXLhATEyM3uf1jwCpVMq2bdsQiUS0t7dz69Yt\nLly4IDih/9TxmvD8RNDX18eXX37JihUrRtya6u7uJiMjY9h05rHAWOZoDRfYOdYYKjT0ZdPRdS7O\n27ZtY+/evSiVSj755BMh9ftl34+zszPW1tbk5+cP+Pu+vj7i4+P53//9XwAOHjxIWFjYkMeqo6Nj\n0KrRgQMHMDY2xtDQEGNjY+bPn09DQwPHjx8XRKaDmQLa2dkRFBTEnTt3hJ/5+Pjg7OzM/fv3qa6u\nBhCCSY8dO8bly5cZP348v/zlLwkLCyMnJ4fx48fj7+/P1q1bhcqasbGxQHgsLS0xNTVFJBKxceNG\nMjIy+OSTT2htbSUqKmrIO3UDAwO2bNmCSCTiyy+/HDHpSUlJoaurix07dgieMboU8vb2dv7yl7+Q\nn5/P3Llz+dWvfsXy5cuxtrbm2rVrQjtr3bp1LF++XKjo6GIPBkNJSQmOjo56ZFxXEZ06dapQgRsI\nuvNQ19bSCZyrq6uFNt9I8PjxY2pqali9ejXe3t7MnDlz0CqvsbExO3bs6OcQrjMe1Ol5hooNGQ0U\nFRWhVCqZNm3asI8Vi8UEBgby/vvv4+joyGeffcbNmzeHbNt93zAyMuLtt99GLBaTkZFBQUEBjx8/\n/qF363vB65bWTwBarZYrV67g4+PT725oIOTk5LB582bq6urYvXv3S0UOjBZ+CNNBHX6oCs93TUfX\nuTiHh4eTlZU1qIvzcNDla02ePFn4mVKpJC0tjaSkJCZOnEhUVNSIzwtd1MNA7zk0NJQ//elPNDY2\nCqRJrVZjbGxMTk4OarWamJgYFi5ciIuLS79js3DhQj7++GPKy8sFga+ZmRkTJkzg3LlzzJgxg2fP\nnmFmZsa8efPw9fUV7vblcjmpqans2bOn3369WOF5EXV1dWg0Gjo7O9mzZ8+IzlGJRMKbb77JV199\nxfnz5wXvn4GgC+YtKipi3rx5uLi40NXVxZMnT3j8+LHQ5o2MjBRuXDQaDY8ePeLevXt0d3fj7OxM\nT09Pv+96VVXVkDcv2dnZeqLjZ8+e0d3dLRgK3r59m/nz5w/6fB8fH549e4aLiwvR0dFMmzaN8PDw\nEVdXdO7Zu3fvFiqcc+fOxcnJiX//939HrVYLrUdTU1NCQkI4dOhQv9fp7e2ltraW6upq7O3tefDg\nARMnThwTUz2tVsuDBw/QarUvJfY3NDQkNDSUoKAg7t27x+HDhwW/qe8aLzEaUKvVqNVq4YYpOztb\nMMD8KeN1hecngOTkZDo6Okacl7J582by8/Npb2/n6NGjVFVVjfEe9sdY5WgNFyvxQ1Z4Risd3cjI\niDlz5vD+++8TGhrK48eP+etf/yosiMPB39+fpqYmGhoaUKvVZGZmcujQIaqrq9mzZ48Q4jkStLS0\nkJeXx4IFCwb8vW4R++Uvf4lUKhViHoqKiqirq0MqldLX10d0dDR///vfSUtL0yMihoaGLF++nJs3\nbwqVILVaTVNTE2q1mtTUVBYsWMDevXvx9/fXW3zT0tLw8fHBwcGh336ZmprqbUetVnPnzh2uX7/O\ntm3bmDlzJteuXRvx+SKRSNi0aROmpqacO3duwGR2jUbD1atXKSkpwcbGBgMDA86fP8/HH39MU1MT\n69at4+DBg9ja2qLRaNBqtTx79owjR47w9OlTlixZAjwfE//2RFBnZydyuXzQ1rRcLqe4uFggue3t\n7Vy/fp358+fT3t6Op6cn3d3d/TLFXoS3tzfFxcWcO3eOVatWvVQrSalUEhMTw5IlS/RG3v39/bG3\nt+dvf/sbUVFRrFy5knXr1vHHP/6RuLi4ftcIlUqFSqXim2++wc/PDw8PD8LCwrh8+fIri8eHQmVl\nJe3t7VhZWb3S9crMzIzVq1ezd+9e6uvrOXz4sGDd8EMiMzOTjo4O4Qajq6trwDbxTw2vfXh+5Cgv\nL+f27dvs2rVrxIm7f/jDH4QSq5GRETt27PjeJ7Rqamro6OjQqzKMBuLj4wkNDR30Lqquro6Ojo4x\nTYLPzs7Gzs6uXyvkyZMnmJiY4OvrOyrb0ZGn6dOnM378eEpKSrhx4wbNzc1YW1sPWukSi8UoFAoy\nMzN5+PAhHR0drF27lnnz5r20J9LNmzeFQMmBUFFRQV1dHUZGRrS3twuxEmq1Wth/ExMT9u3bh52d\nHQUFBdy4cYOGhgaMjY2xtrbGwcGBZ8+e0dLSQm5uLvn5+Zibm7N161bGjRvH7du38fb21iOZcrmc\nixcvsmHDhgHfU2NjI62trfj6+tLR0cH58+fp7e1lx44dODg4MG7cOFJTU1Gr1SMWn4pEInx9famo\nqCAtLY3JkycL7RqlUkl0dDQdHR24ublRUFAgtEnWr1/PlClThAymgoICgYCVlZWxbNkywsPDycjI\nQCaT0d3dzdq1a/UW4NLSUrq7uwet8Obl5aFQKJg5cyYajYYLFy4wbdo0Jk2aRGZmJrNnz6ajo4Pm\n5uYBBfUKhYJbt27R1NTE+vXrX9rH6vr165iamhIWFqZXxbOysuL27dvI5XJ+97vfUVNTg6urK729\nvTg5OfUjq7o2/MSJE9Fqtbi4uDBr1izy8vLo6OgYdZf4GzduCPlY3+WaYWpqypQpU/Dw8CApKYnU\n1FRsbGywsbH5QUwAx40bJ1RDGxsbUavVlJSU4O3tjUQiGRMvsH8EvK7w/IjR2dnJxYsX2bhxI1ZW\nViN+3n/+539ibGyMVCpl9uzZJCQkcP/+fcrKyujr6xvDPf4/jEWFR6lUolKphnzdgaZMRhuDtbQG\nGkcfLehcnD/44ANsbW05e/YsJ06cID8/X+9uUqvVUlRURF5eHuXl5Sxbtoxdu3a9kjePTCajtLR0\nyBZdQUEBkyZNIjs7G41GIyy4ugvq7NmzhQqjj48PERERfPjhh7i6unLr1i0OHTrE119/jVKp5OHD\nhxgZGTF+/HiCgoKws7PDz8+PNWvWcObMGerr64XtpqenD1rdgf9raZWUlPDpp58yYcIE3n77bYEc\nSSQSIiIiuHfvnt7rDgexWMz69euxt7fn9OnT9PX10dbWxpEjR6iurqa1tRWlUomlpSV79+7tZ+ZX\nX19PQ0MDycnJhISE8O677wrnTG5uLk5OTri7u/f7vg+n38nJyRHaWQ8fPkQkEjF//nxsbGxoa2tD\no9Ewbdo0wcH5RTQ3N/P5558jEokIDAwUtDMjRXZ2tuCqPZDw28LCAhcXF8rKygTTQYAvv/xS0Gnp\nUFZWhlqtZvny5UKGlkgkYv369aSmpr7UsRoO9fX11NXVIRKJRs2oz93dnT179nsAeeMAACAASURB\nVBAWFsatW7c4deoUdXV1o/LaLwtnZ2dWrVrFv/7rv7JmzRrOnz+Pu7s7jo6OZGZm/iD7NNb4adK4\nnwF02odZs2a9lAgW4J133mHZsmX813/9F7t37+bWrVvcvXsXQ0NDVCoVVlZWvPvuu4Pa2o8GxsqD\nx9zc/Ae3TR+opfV9paObmpqycOFC5s2bR15eHg8fPuT27duEhITg4ODAw4cP6e7uJiwsjLy8PNra\n2l55W/Hx8SxYsGDQ80Sr1VJYWMjy5ctJS0tDpVIxceJEPD09sba2JjAwkPPnzyOVSgUnZd17CAkJ\nwcLCgnv37pGbm4tarcbGxoa6ujpMTEz0FmV/f3+0Wi2nT59m586d2NjYkJycPKB2RwcjIyNBBxIR\nETGgQaetrS0rVqwgJiaGqKioEUdIiEQi1qxZQ3R0NP/zP/+DXC7HxsaGdevWMXHiRPLy8vrdWLS0\ntHD37l3KyspwcXHByclJr1pTU1ODQqFAqVQKIawvQhcYOhC6u7uprKzkzTffpKqqirS0NKKiohCL\nxYjFYkxNTeno6BA0VLW1tUJV69mzZ3z99deEhYURFBREcXExSUlJI7Y1aG5u5tatW+zcuXPA80Tn\npm1qakpsbGw/snX8+HE++OADrK2tUavVJCQkYGNjg5GREY2NjcI5Y2VlxfLly/nqq6/Yv3//qFQo\nEhMTmTNnDo8fPx7VsXeRSISfnx+TJk3i0aNHnD17Fm9vb9rb28nLy+Nf/uVfBiXqYwGdsWRGRgZK\npRKlUsmf//xnLl68+L3tw/eF14TnR4rY2FhBGPcq8PDwYNKkSZSXl7Nw4UIePHiAQqFALBZjb28/\n5gFzfX19oxopAcPrd+CHq/CMdBx9tCCRSAQX56dPnxIbG0tnZyfjxo1jy5YtODg4YG5uzpUrV0Y8\nTvwiqqqqkMlkbN68edDH6HQ2tbW1uLq6As+nmnbu3Clsb/PmzZw+fZonT56wfPlyFAoFjx8/JiUl\nBTMzM8LCwvD19UWlUvHkyRNu3bolpKW7uLhga2sLwOTJk9FoNJw+fZqpU6fi7e096KLR1dVFfHw8\nPT09fPjhh0OK3AMCAigtLeXmzZv9YjAGQmdnJ0+ePBF0GgqFAjMzM/bu3Su0nF9M9+7q6uL+/fvk\n5OQwe/Zs1q5dS25ubj9dXVpaGgYGBvT09PQLnVWr1dTV1Q3aesvLy2PixIloNBouXbrE2rVr9c5P\nGxsbWltbsba2ZurUqWRnZ+Pi4kJcXBy5ubls375deG0vLy9iYmJQKBTDnsu6cf2wsLB+uXI6PH36\nFC8vL8rLywec1tONvkdERJCUlISRkRFWVlY0NjZibW2ttw8BAQHk5+eTkJAworDkodDa2kpJSQlL\nly7l3r17Y9Ly17mfBwQEEB0dTVRUlGCtcOHChVHf3lAwNzfHxMQEpVKJsbHxmFWif2i8Jjw/QuTl\n5ZGfn09UVNR3Wrzd3d2prKxk3bp1JCUlCUJJe3v7MScGvb29oz5VMZIJrbGGSqXSI3OlpaUYGBhQ\nWFj40pW474rW1lbu3r1LaWkpCxYsYNKkSWRlZXHixAnc3NwICQnB0NCQkpKSl9o3rVZLXFwcixYt\nGvJOurCwkEmTJpGTk4Ojo6NQ3XrxvBo3bhyzZs0iNTWVvr4+CgoK8PT05I033tAbdTY0NCQ4OBgT\nExOuXbuGSqXi888/x9HRkcDAQPz9/Zk6dSpKpZJr164NGoJbXl7OpUuX8Pf3H/H5snr1ao4eParX\nFnoRarWawsJCHj9+TGVlJf7+/kKr+I033qCuro6TJ0+yc+dOzMzMqK+vJygoiLt375Kenk5AQAAH\nDx4UzhmdF8+Ln3dBQQHW1tZ6U2g61NfXC1WPgZCTk8O8efO4fv0648eP76dFsbW1paWlBW9vb6ZN\nm8aJEycEQXlUVJSeNtDQ0BAXFxcqKyuHPWdu3bqFvb29Xo7ai9BqtaSlpbFq1So6Ojro6upCLpcL\no/0ikYi9e/fi5uZGc3MzycnJzJ07l9bWVj3SqINIJGLt2rV88skn+Pr6jnhUfiAkJiYSFBREU1PT\noNlqowG1Wk1LSwuWlpYYGRmh0Wh+EMNCY2Nj4uPj+d3vfkdAQAA/VWnva8LzI0NTUxPXr1/n7bff\n/s5j3b6+vuTk5CCVSgkKCiIjI4OdO3eSkJDAmTNniIiIGLEQ+mUhl8u/9xwtGPsKj24fdNuIi4tD\nJpOhVqvx9PTEwsJiyDyi0dqH+/fvk5ubS0hICGvWrBEWwxddnO/cuUNPTw937tzBy8trxOOyJSUl\ndHV1CQnbg6GwsFBYoGtra1m2bFm/xzQ1NdHV1YVGoyE7O5stW7YMeXc5ZcoUbt26hZmZGb/+9a8p\nKCjg0aNH3Lp1i6lTpyIWi3F1deX69evY2dkJpFqr1fLw4UNSU1PZuHEjnp6eZGVljej9GhoaEhER\nwenTp3FzcxMCYRsbG8nKyuLp06fY2dkRGBhIREQERUVF3Lhxg82bN+Pt7S1YBnzxxRds27aNqqoq\n6uvrBx3//zbhqaioQKlU0tHRQWBgYL/9G0q/097eTmNjI93d3chkMvbv39/vMTrCA889bnp7e7G2\ntmbDhg0DLvTe3t6UlpYOSXhyc3MpLS3VS13/NsrLy4HnpHfu3Lnk5eVhaWmJg4MDAQEB/Nd//ZfQ\nvvz6669ZuHAharUaExMTIZbj29BNRV2+fJl33333lSqqXV1d5Obm8v7775OZmTnqBKSzs5Nbt25R\nW1tLR0eH0MbLzMykuLiYlStXjur2Roru7m5CQkL47W9/+71Vor9vvBYt/4igUCj48ssvCQ8PF1oE\n3wU63UJNTQ1hYWFERUXh6enJjh07cHFx4ejRo9TU1Hzn7QyEsTAeHEmO1ljj2+0sNzc3oVRfWVlJ\nQkLCmI3G9/b2Ehsby//+7/9iYGDAwYMHWbx4cb87f52L84EDB1i/fj3Nzc385S9/GZGLs1arJT4+\nnrCwsCHvent6epDJZDQ3N+Pj44NEIhHaT1qtloqKCs6dO8eJEyewt7fHzMyM8PBwrl27NqQ7rUgk\nws3Njfz8fPr6+pg8eTI7duwgKioKIyMj0tLSkMvleHt7c/LkSZqbm+np6eHs2bMUFRURFRXFhAkT\nkEqlaDSaERsFuri4sHDhQqKjo0lPT+ezzz7j1KlTSCQSIiMjiYyMZMaMGWRlZXH79m127typV9Fa\nvHgxdnZ2HDp0CLVazc6dOwcd/7ewsNAz00tOTsbIyAhXV1eBbL2I6urqQasZOTk5eHt7C22hgUit\njY0NLS0tJCcn8+WXXxIQEIBEIhn0+I4kV0tH+IbSAaampgrtVF9fX4EAaTQajI2NMTU1JT09nays\nLBQKBbNnz6anpwdTU1Pq6+txcXEZ8HX9/f1xd3cnNjZ20G0PhZSUFKZNm4aZmdmwYvBXgUgkori4\nWBCLi0QiFi1ahK+vL2vWrPlBktdbWlq4f/++sG8/VbwmPD8SaLVarl27hpub24CixVeBmZkZUqmU\n/Px8DA0NBc2DWCxm6dKlrFixgrNnz5KZmTnqi/RYGA+OpEUx1hWebwuW3dzchAuYmZkZu3btGvXt\n6yaYDh8+TE9PDwcOHGDFihXDaqR0dvghISH4+/ujUqn45JNPiI6OHtTF+dmzZwDD2gkUFRUxbtw4\n8vPzMTIyYsKECWi1WnJzc/n888+5evUqEydO5J/+6Z9YvHgxXl5emJiYEBoayqlTp4YkXiYmJnh4\neOgtaNbW1hgbG+Pn58eKFStQqVT09vZy5MgRDh8+jKOjI7t37xbI6LfztIaCVqulvLycuro6ZDIZ\nKSkpLFq0iF/+8pd4e3tjbW2NVqslNjaW9PR03nnnHaH6oGtH/f3vf6e7uxsfHx/EYvGQ576xsbGQ\n3q7RaCgtLcXQ0HDQ1lBVVdWgi/LTp0+RyWSEhoYOqkMxNzenrKyM7Oxs9u7dy+LFi8nPzx+UDLq5\nudHW1jag55NKpSImJobQ0NBBCQk8b7dWVlYK7sW6nKr29nbhBsHd3Z3CwkLi4uJYt24dYrFYuFGS\nyWSD6oIAVq1aRUFBwZDEbCD09fXx6NEj5s2bh1arpaamZtQJj5mZmZ5rs4GBAXPnzh3VbbwMdDYF\nOiPCJ0+e/GD7MtZ4TXh+JEhPT6exsZHVq1eP6oLp4OAw6EXB39+fyMhIUlNTuXr1KkqlctS2OxZT\nWiMRLY81vr0Pzs7OqNVqpFIpkZGRg2ZsvQrUajXp6en87W9/o66ujsjISNavX/9SFgXw3Hk5Pz+f\nJUuW8Mtf/hJPT0+++uorPvvsM54+fSosQBqNhvj4eMLDw4c9BwsLC7GyssLe3p7q6mo0Gg2HDh0S\njAIPHjzIrFmzhIqDp6cnlZWVzJo1i5CQEE6ePDlomrOOqJWUlAhjywqFguTkZBYtWsSECRN48803\nBdfgvr4+8vLyhDgHHYYjPB0dHTx48IBDhw5x48YNnJ2dOXDgAEqlErFYTFtbG6dOneLSpUtcvnyZ\n8vJy3nnnHaFqU1lZyfHjx4mLiyM8PJzIyEg8PDxwd3fnxIkTg07IiUQiocpTUlIimO0N5N/U2dmJ\nQqEY0HCwsbGRtrY2bG1tCQkJGXBbjY2Nwnc7MjISGxsbrKyscHBwoKSkZMDn6KZ6Brpu3LlzB2tr\n60G3p0N6enq/kXx/f39aWloEG4UZM2bQ0tJCYGCgQG56e3sFp+6h2u3GxsasX7+eK1euvJTVhs7j\nx9ramra2NiQSyah+Z/v6+oiJiaGmpkZoT86fP/8HbSElJibqGU7qzrmfIl5reH4EqKqq4t69e+zd\nu3fUbcm9vb1JTk5Go9EMWMK2t7dn3759XL16lePHj7Nly5ZRiaIYy7H0ofB9VHgsLS1RKBRCO1Ai\nkfD222+PWkCrVqslOzubhIQEbG1t2bZt23dqcdrY2ODh4UF2djYzZ85k9uzZBAcHU1RURGpqKnfu\n3GHWrFkYGRlhbm4uZD8NBp2Jmbu7OwYGBkKFYtOmTYO2Xry8vEhPTweexw0oFApOnTrF7t27+y1s\nIpEIAwMDli5dyo0bN9i3bx8ZGRl4eXnh5OREb28vV65coauri/fff5/CwkLu379PbW0tDx8+/P/s\nvXdU1He+//+YQq8DiMCASIkgIE3sBUGN0VhjSaKxxJbdbO7u3Xv37H633HPvPffs3puzvWQ3G91o\nNKJRUFFELGADKSowogiK9DIgMMDQhmm/Pzjz+TFSRMUkm/g8x4OHGaZ+Pp/38/16PV/PJxMnTiQ6\nOtosT2vgay8tLaWgoIDa2lpCQ0NZu3YtXl5ewnGzZs0ajh8/TnBwMGKxmOLiYqysrPjhD3+IlZUV\nTU1NgnZrwYIFhIeHC+dWY2MjU6dOpbu7m/3797Nlyxah1TcQJh1PdnY21tbWREZGDtnqMLVchjqm\nr169itFoZNWqVUPefvfuXVJTU1m0aBHp6en09vYK1xfTtNZwJpmmttbASsW9e/e4f//+iLodQJjE\n2717t9nvg4KCSE1NFTZWpscYmFDe09NDT0/PiNUdEwICApg0aRJpaWmsXr36iffXarXk5OSwefNm\n4MneRk+LhoYGjh07RkBAAGvWrEEsFuPk5PTCk96fBH9/f6GyJZFI0Gg0tLS0fOlmtF8GXhKerzm6\nurpITExk5cqVQ14Ynxe+vr6CYddwi6ZJtJmTk8PevXtZvXr1c00cGY1GNBrNmPv8fB2mtBobGzl8\n+DDLli1Dr9djMBhwcnLC1taW73//+8/lD2IyDczIyEAqlbJixYox8/WZNm0a6enpREVFIRKJEIvF\nBAUFERQUJLRxCgsLeeWVV57YTlAoFEgkEsG5VS6X8/bbb4/4/O7u7nR2dgrf4fz58+nr6+Pzzz9n\ny5YtZuRYJBJhNBqZMmUKt27dEjLANm/eTH19PceOHSMoKIj169cjkUiEakN2djbvvvuusIFobGwk\nNzdXIKgFBQUUFRXh7u5OZGQkGzZsQCqVCu0eLy8v3Nzc8PPzIzIykqysLKHtp9FoOH36NFKplLKy\nMubMmcP69esHfd8NDQ3Ex8fj5uaGWCzms88+Y8uWLYPIsKOjI21tbVRVVSGVSodtYw/XzjIJb5cu\nXTqotWlycS4tLRX0evn5+ahUKqE6GRoaSnp6+rDj5/7+/sL7F4lEqFQqUlJS2Lhx4xM3MqZR9Mc3\nTnZ2djg4ONDc3IxGo+Hs2bPY2dlRVFQkjOJ3d3fT3t4+KsIDsHjxYj7++GNKSkqe6JRcWFiIXC4X\nFvra2toxESwbjUZu3rzJ5cuXWbp0qdnQwlcR3Pw45HI5crmc27dvs2vXLhwcHL5yL7MXhZeE52sI\nlUqFtbU1VlZWJCUlER4ePmZxBI/D09MTg8FAZWXliFUCkUjErFmz8PLyIjExkWnTpjFv3rxnOjE0\nGg2WlpZjOuppMBjo6el5om7lRVZ42tvb+fGPf4xKpTLLUurp6eE//uM/uHDhAikpKc8kSqyqqhJ2\n4fHx8QQFBY3p+wgICODs2bNDCmDHjx/P+PHjCQgIwMfHh4SEBFxcXJgxY4YwhWXyD7l+/Trl5eU4\nOzsLItvRtNjEYrHQ1goJCUEkEgkVnMOHD7Np0yZh4TURHpFIxNKlS/n000+ZOHEiVVVVXLlyhddf\nf32Qxmj69OkYDAaOHDnCtm3bmDp1KgkJCTQ1NfGXv/xFaJNt27aNcePGYTAY2Lt3L//7v/9LY2Mj\nFhYWaDQa5s2bx//8z//g6Og4SON09+5d5s6dywcffDDkot/b20t3d7ewcYmJiRGmtzZv3mzmG+Tg\n4EB5eTkGgwEPD49hq4O1tbUsWLDA7HdGo5Fjx45hZWVFTEyM2W0dHR0kJiZibW3N7t27BS2RaVLL\nVE2xtbXFx8eH0tLSIVPCTa+ntbUVZ2dnEhMTmTt37hMJwsBR9KHg7u5Oc3Mz6enpgti9tLRUuL2n\np4fW1tZRXw8tLS1ZvXo1x44dw8fHZ9jrg8Fg4Pr167zxxhvC7+rq6p46QuNxmIhwS0sL27dvH7Mq\n71hDo9Gg0Wi+0WQHXhKeryWSkpJQqVTCeG5cXNwLey47OzusrKwoKytj9uzZT7y/r68vu3fv5tix\nY9TV1bFmzZqnbk29qAktW1vbF+aXMRr84Ac/oKWlZcj+d3d3N1evXuXjjz/me9/73qgfU6lUkp6e\nTnNzMwsWLGDKlCkv5D2KRCKmTZtGXl7eIMKj0WjIzMxk8+bNjB8/ntmzZ3Pv3j2uX7/O+fPnmTBh\nAk1NTfT19TFz5kwaGxuxsrIiKiqKixcvDuuJ8zgGEh7Ta1q2bBnJycl88cUXvP3220ilUoHwQP/C\nazQaUSqVqNVqduzYMWwldObMmej1evbu3YtcLqe8vBxXV1c2bNggtFn2799PaGgoH3/8MRkZGYIw\n19T6unjxIllZWfzoRz/Czc0NlUolPL5pEm24Y1upVOLu7m72/UVFRSGRSDhw4IBZpp2joyOFhYVY\nWloOIi0m6PV6lErlIJJx48YNmpubmT59utniVVFRwfHjx4fcrAwcTTfBFDUxFOERiURCW6u1tRUH\nB4dRtWYqKioQiURDOltDP7kuKyuju7ub733ve7S3t3Pr1i20Wq1gvNjc3DzqCg/0H1fh4eGcOXOG\n9evXD9vec3R0FI59nU5HU1PTc7WKlUolx44dw8/Pjx07dnyt86mam5txdXX9RpMdeCla/trBaDTS\n2NhId3c3CoWCgICAF34Qenl5UVtbO+pJLAcHB7Zu3YqzszOffPLJU6fsflUTWi8S7e3tfPHFFyOK\n/bq7u/nwww9H9Tm3tLSQlJTEoUOHCAwM5Hvf+x4REREvlNBFRkZSVlZmNhIN/WO6/v7+wmIskUiY\nNGkSYWFh6HQ6ysrKaG1tJSAgACcnJ4xGI83Nzbi5uaHT6czSsUeCifAMhCknydramsTERPR6vVBR\nArh06RIikYienh6WLl06LNlpb2/nypUr3Lx5U6hGRUdH4+vrS1BQEFOmTGHz5s3s3LmTs2fPkpaW\nNmzyfE9PD7/97W+FAF7oF8mapoeGw1BmedDvELxkyRIOHjwoZEHZ2toKzz9clWEow8HGxkauXLki\nPC70X1OysrKE3L358+cPuqaY3JYHIigoiOrqarP3ORB+fn4oFAru3bs3rE7oceTl5Q0iYgNhIosR\nERHY2Njg4eGBRCKhqKiIvr4+pFIpBoPhqYXEcXFxNDc3U1RUNOg2k0fT3Llzhd8plUpcXFyeSUxs\namEdPHiQBQsWsHz58q812YH+683Xtfo0lvh6fwvfQjw+uXHx4kWcnJxeqFndhAkTqK2tNcszehIk\nEglLly5FLpdz4MABlixZMmxS8+P4Kie0XlRLKz8/H0tLyydOhCiVSjIyMnBxcRG0MgN/miaKamtr\nCQoKYvny5VhaWlJXVzfk/Z/350BYW1sTGhpKfn6+oC3o7u4mNzeXnTt3Av3EMi8vj1u3buHr68uG\nDRvw8fGho6ODmzdvkpiYiEQiwdvbm4qKiqci7F5eXjx69IiGhgZaWloIDg5GKpUiFot54403+OKL\nLzh58iS2trYYDAby8vLIzs4mPj4eBwcHzp8/z44dO4Tn0+l0lJSUUFhYSH19PaGhoaxfvx5PT08y\nMzPJyspCo9HQ1dXFokWLkMlkODs7c/z48SdOJOp0OlQqFR9++CEpKSn09fWxZs2aEd+rUqk0E+AO\nhMk08fPPP2fjxo00NjbS19fH9OnTh10sH9fvaLVakpKSiIyMpLKyEldXV3p7e0lOTkatVrNr165h\n24tDVXhMdgL37t0bciTezc2Nuro63n333VFtYFQqFTU1Naxdu3bY+1RWVmJpaWlGNt3d3SkqKsLP\nzw8LCwshMPRpIJVKWbNmDZ9//jkTJ040I0xlZWWIRCIzXeKzCpY1Gg0pKSk0NTXx7rvvjrmb/IuC\nqcLzTcdLwvM1Q01NDTqdDolEgqWlJbGxsU8U2z0vPD09kUqlVFZWPrUyPzw8nPHjxwvJxkuWLHmi\nRuWrmtB6Uejp6RG+t9Ggvr5ecBc2Go2CAZ5pjNjJyQkvLy86OzvJzc01u99Y/gQGkSCj0YhWqyU3\nN1eY2BCJROzfv5++vj76+vqwsrLC1tYWlUrF2bNnzR5DJBKh0WioqqqivLwcBwcHDh06JLSiHidc\npv+r1WoaGxvR6XTs2bMHo9HI1KlTsbGxEe7r6elJUVERGo1GeG2urq5YW1uj1Wrp7u4mOTkZmUxG\ndXU1tbW1ODs7CyJjCwsLurq6ePjwIXK5HAcHBzQaDcXFxZSWluLn54evr+8TzRehf9IoISGBDz/8\nkCVLlrBnzx4UCgWRkZHD/o1SqRxxXDskJASxWMzBgwfJzc3lxIkT7N+/f9j719bWmi3S58+fZ/z4\n8XR1dREWFkZjYyNHjx4VUuhHqjK4uLjQ0tIyaEMQFhZGTk7OIMKj1+sFUfFoq455eXnC9zAUmpub\nqaysZOLEidy/f1+YHA0KCiI7O5uenh7EYvFTtbMGwtPTk+nTp3Pq1Ck2bdokvM/MzEzmzJlj9r7r\n6urw9/d/qsdvbGzk2LFjTJgwgZ07d475RO2LhGmD8U3HS8LzNYHRaCQ/Px+FQoFUKmXp0qVERER8\nKa6bXl5e9Pb2UllZyYwZM57678ePH8+uXbuEC/T69etHLDm/KA3PaAjP81Z4DAYDTU1N1NbWCv9M\n1aXR+BTJZDI2bdokLBJ9fX3k5OSQk5PD5MmTiY2NHVPfj5EwHBk6fPgwEREReHh48NlnnzF+/Hia\nm5uJiIggNDRUMMYb+HdGo5Guri6SkpKwsbFhxYoVHDt2DDs7O2pqapg4cSL+/v7Y2NgM+Zx9fX1C\nnILRaDTzPxl4Px8fH27fvi04AXt4eNDY2IhWq8XCwkI4f1xcXIS4jPb2dsHVduDzDhSW6/V6ysrK\nqKioGHX7QaVSUVZWhr+/P+vWreOzzz7D29t7yF29TqejpaVlUHtPo9FQV1dHTU0NtbW11NXVIRaL\nsbCwwNraesSK4UDBcklJCWVlZezYsYOPPvoIT09PDhw4wKuvvvrECBBAqND09PSY2QAEBgZy6tQp\nwW7BhIyMDKytrQkLC6O8vPyJ1ZC+vj4UCsWgUXQTTPERkyZNwsLCAmdnZyorK/H39ycmJobLly/T\n0tIiiLifFXPnzuXTTz/l1q1bxMTEUF1djVqtJjQ01Ox+dXV1ow5mNhqNFBQUkJ6e/lSV7q8TXra0\nXuJLxc9//nN+97vf4eDgQEVFxZdarbC1tcXGxobKyspnJgTW1ta89dZbXLt2jT179rB27dphhYkv\nqqU1Wq3I06Czs1NYkOrq6qivr8fR0RFvb2+8vb2ZNWsW48aNQywWc+LECa5cuTJk6jP0Lyr/9m//\nhlgsRqfTcevWLcEXZseOHV/6BUckEg1JqE3Bl93d3UgkEiIiIggPD3/ijrWwsBBHR0dCQkKwsLBA\nLpezfft2WltbycvLIyMjA39/f2bMmIGPj4/ZcWYyITQZHU6cOHHQgqNQKMjPz8fPz4/6+nrs7OyI\nioqisLCQBw8e8Morr+Dk5ISjoyPLly9/4vu/cuUKly9fRiQSYWFhwcaNGxGJRPzyl78c1ednY2PD\nkSNHEIvF+Pr6EhISQmJiIjt37hxEmh49eiToZAaSZZVKhaenJ97e3kRHR7Ny5UqKioro7e0lOjoa\nlUo1JJkfaDjY0dFBSkoKb775JpWVlVhYWJCXl8eWLVtGXbEViUS4uLigUqnMCI9UKhUy90xDDQ8e\nPODOnTu899571NXVcf369SeSA4VCMeQougn5+fnodDoCAwOpqKhg8uTJ3Lt3D39/f+zs7LC2tqa4\nuBitVjuig/OTIJFIWL16Nfv378ff35/MzExmz55tVqXq6uqiu7t7VO2ovr4+zpw5Q0NDgzDh988G\no9FIa2vrS8LzEl8e8vLy6Ovro62tjczMTCZMmICFhQVSqRSpVIqHh8cLKGSnDgAAIABJREFULZF6\ne3tTVVVFc3PzM5+0IpGI+fPnI5fLSUxMZPbs2cyaNWsQgXpRLa0nGeLByBUe09TLwAWpt7cXuVyO\nt7c3c+bMQS6XC1WKlpYWGhoauHnzJqWlpcyePRuFQkF7e/ug9pa1tTUhISG8//77KBQKLl++jJub\nGxs3bnyuC/hYwuRFk52djVqtRiwW86//+q9PHPU3obS0lO7ubsLDw8nPzxe+DxcXF1577TXi4uIo\nLCzk5MmT2NjYMGPGDCFYE2D58uU0NTVRV1dn5i+k1Wo5e/Ys1dXVbNmyhdzcXKqrq1GpVJw8eZJ5\n8+axbNkywTn5o48+YurUqU/8XMePH4+FhQW7du1CoVCQlpbGli1bCAkJIT8/f8S/tbOz47/+679Y\nu3YtxcXFlJSU8PDhQ4xGI7/5zW8IDQ3Fw8NDaNcVFxfT2tpKQkKCQJanTp3K+PHjB5HOW7du4efn\nx5w5c0hISCApKYmtW7ea3cek3zEajZw4cYJp06bh6OhIQkICzs7ObN269anPMZOO5/GprylTpnDx\n4kVmz55NR0cHycnJrF+/HltbW3x9fUlMTBzWrwf+/1H0119/fcjb1Wo1GRkZbNmyhaamJgwGAyEh\nIezfv19wlpfL5dTW1qLX65/bj2zcuHHMnTuXY8eO0dnZSWBgoFmFo66uDrlc/sSNX1NTE8eOHcPb\n25tdu3b9U7WwBqKjowMrK6sx90X7OuIl4fka4NGjR8ycORNnZ2csLS0pKCggNzcXqVSKRCKhr6+P\nVatWjao0/azw9PSkubmZqqqq596lBAQEsHPnTo4ePUpdXR0rV640O5l6enrGfCf0LBqejo4OM3Jj\nmsyQy+X4+/sTGxs77KimyYHYwsJCaI14eXlx6NAhDhw4wPHjx7GyshJaKLt378bDw4Pf/e532NnZ\nsWjRohH1Hl8m1Go1eXl55Ofn4+vry7p16zh9+jQSiWTUZEen0/Hw4UNkMpkQS7Bq1Sqz+1hZWTFj\nxgymT5/OgwcPyMnJEVycY2JisLOz45133uGPf/yjQFZaWlo4duwYrq6uzJkzh3PnzlFVVYWlpSXr\n1q0jNTUVg8EgtGRsbGyIj48nNTWV7du3j7homUjVuHHjWLhwIQaDgX379jFnzhzu3Llj1vIaCFMO\n15YtW3BwcMDDw4O4uDgaGhrIzs7mzp07AmEykRkLCws8PT2ZNWsWcrkcR0fHIV9bd3c3KpWK5cuX\nM3HiRN566y0OHTrE1atXzaooJlHt9evXMRgMyOVy9uzZg1arZfPmzc+0oRhKuAz9IcNqtZpHjx6R\nkpLCjBkz8PX1Bfp9bjw9Pamurh7WjLS8vFyogg2FtLQ0oqOjhdapXq/H1dUVW1tbampqhLHykydP\nYm9vPyaTijNnzuTatWvo9XrS0tJYuHChEEUyGsPBgoICLl68OOqW4dcZponKbwNeEp6vGJ2dnSQk\nJLBu3ToiIyPJzs7m7t27KJVKIUPHyspqUI95rOHl5YVCoaCqqmpY34+ngbOzM9u3byc1NZW9e/fy\n5ptvCieVRqP50qe0tFotDQ0NVFZWolKpuHPnDnq9Xthtx8XF4eXlNepdTlRUFFlZWcI0iaWlJfb2\n9kycOJFDhw7R1tbGnj17mDx5MoGBgWRlZQli2K6uLs6cOUNWVharV68eEzfXp4VpdPz69euUlJQw\nZcoUwcNGqVTS2dmJXq+np6dnVBM4pumayMhI2tvb6erqGtbDRCQSMWnSJCZNmkRTUxO5ubn85S9/\nITg4GDc3N/72t7/xs5/9jNmzZ/Paa68hl8upqKhAo9EQERFBXV0doaGhBAQEsHnzZvbv34+lpaXg\nRhwVFcWtW7eeKCK2tLREr9cLQwKmlpO/vz+/+c1v+MlPfoJWqzWr1llaWuLg4MB//dd/UVlZSUtL\ni6C9sbGxwdvbm5kzZ1JQUEBYWBilpaU4OjqiVqtxdnamsLCQM2fOIBaLkcvleHl5CT9tbGzIyclB\nKpUil8v59a9/zcKFC1m8eDEXLlzA1dVVuA7U1tYSFhbGlStXiIyMJDk5mcjISJqbm0fMmBoJMpmM\nqqqqQb8Xi8WEhoaSkpKChYWF2fg29BPH8vLyYQlPXl6ekIr+OEpLS1EqlaxZswboJ4imLK3JkydT\nXFzMhAkTCAkJ4cSJE8/83h5HVlaWWazIwCypurq6YbWMfX19pKamUl9f/0/bwnoc3xb9DrwkPF8p\n+vr6BIGo6cI8Y8YMFAoFwcHBwqSCRCLh8OHDLFy48LmMsEaCp6cnarUajUYzZqPbUqmUlStXkp+f\nz759+wQH3LEWLZsEs6YKj9FopK2tzax68+jRI0FrY2qxyGSyZ36fFRUV6PV6YQc/Z84ciouLBWGn\nXq+nu7ubu3fvUl5ezoIFC/Dz8+OPf/yjsMi2t7d/Jf4cr7/+OhkZGfz0pz8lNjaWf/mXfzFbSC5d\nusS8efOor6+nsLBwVEnO9+7dQ6PREBYWxv3790c9ju7u7s6KFStYuHAht27d4p133qG6uhqj0UhO\nTo7Qfli9ejXOzs7k5ubi4OAgfNfOzs5s3ryZzz77DAsLC6ZMmSIYFh45coTg4OBhjzVTpaa1tZX0\n9HTUajW7d+/m5s2b1NfXk52dzUcffURSUhI9PT04OzsTERFBbGwszc3NHD9+HJlMRmRkJKtWrTIj\n3DY2NpSXl/ODH/yAsrIyjh49SllZGSEhIbz99tvY2dlRX19PXV0d165do6GhAXt7e9RqNS4uLty+\nfRutVsu5c+eYPHkyfn5+JCcn09DQwMWLF8nOzmb+/Pn4+PhQU1PDrl27OHXq1HNVDV1cXCgoKBjy\nNplMxo0bN/jhD3846Hv19/fnzJkzQ/5da2srtbW1rFu3btBtGo2G1NRU1qxZI5wHYrFYIDwhISEk\nJCSwZMkSYcpvrAKM8/PzzQwsTYTHlJA+1Cbk0aNHHDt2DC8vL3bu3PmVBn6OJb4tI+nwkvB8ZTAY\nDBw/fhw3NzezPBWxWMzy5cs5cuQIbm5u9PT0CLqPw4cP4+vrS3x8/JjnapmEyyZ/kbF8/OjoaDw8\nPIQW12irBqNFe3s7EomE7Oxs6urqqK2tRSwWC9Wb0NBQPD09sbCw4Nq1a2g0mmd+fwaDgYsXL1Jc\nXMw777zDo0ePSEtLY8aMGeTk5AhZQJ999hnQf1HfuXOncEFxcnISDN7mzJnzlQT0tba2IhaLee+9\n9wY9f3V1NY2Njaxfvx6lUsnx48eZOXPmiOTFaDRy7949PDw8cHBw4OHDh08dhWJra8vcuXOFiS/o\nJ40dHR1UVVWh0Whwd3fn2rVrBAUFmZk3urq68s4773DgwAEsLCwIDg5GLpczadIkLl26NGyMASBE\nO0RFRbFhwwYkEgmxsbGcPHmSjIwM5syZg7+/vxC1YJpWamxspLGxEZVKRWZmJkVFRbz//vvC5zR3\n7lwqKirIysoiLCwMJycntm/fTmFhIcePH8fCwoLo6GjmzZsnaMKqqqo4cOAALi4uXL16VagsFRUV\nYWtrS2dnJytXrhQ2JTdu3BCqpz09PdTW1o7a1XooDNfSUqvVXLt2DVtbW9Rq9aDWsVwup729fci2\nsikVfSh9y8WLFwkICDAbbhhIeMaNG4dUKqW+vl7Q1Axngvg06OzspKenh7Vr15Kenk5bW5tAeEwV\nssdbuQqFgvPnz7No0SIh5fybgtbWVsHV/5uOl4TnK8L58+fRaDRDWp17e3sTHByMXq9nwYIFWFhY\nEBMTQ3h4uBDgGRYWxvz588d0msvLywu1Wk1VVdWYEyovLy92795NUlKSIEx8FhiNRqGNYPrX0tIC\n9OsfpkyZwtKlS4fVSDwP1Go1SUlJWFhYsHv3bmxtbZHL5YIpZG9vLxkZGdy7d89M/3H9+nVWrFgB\n9O9ai4qKWL16NUlJSYwfP/5L9b/o6upi9erVXLhwYdCxYzQaycjIIDY2VmirWFtbU1ZWNuIF0eSf\nM3XqVPR6PRUVFcMKVIdDXV2doKVISEjAwsICiURCXFwcFRUVVFVVCWSooKAAe3t7ITXb09MTd3d3\nNm7cKHj+BAYGsnDhQj766CNBHzIQWq2WCxcu0NXVxcyZM5HJZKSkpFBbW0tHR4fQYurs7OQ73/mO\nMF2kUqnYs2cP06ZN49GjRxgMBgwGA1OnTjU73sRiMWvWrOGTTz4BEMjgvHnzmDt3LpWVlRQUFHDp\n0iVeeeUVoqKiKCkpwcbGhg0bNpCQkIBarRZaPAO9nkznjsFgoLOzE4lEQnFxMUFBQc8lnLW3t6ev\nr88s2Ne0MZs2bRp6vZ6ioqJBYnCTPqeiosIshsI0iv7ee+8Neq6amhpKSkp4//33Bz2WacpRJBIJ\n01ru7u4YDAZ6e3vNKqvPgtzcXMLCwggNDWXy5Mnk5uaSkZHBr371K86cOYO3tzebN29GJpOh1WpJ\nTU2ltraWrVu3vpBJ0K8aLys8L/FCkZOTw8OHD9m+ffuwJ+7ChQv561//SkxMjOCOamlpyfz585k6\ndSrXrl3jr3/9K9OmTWP27NljorD39PSkt7eXqqqqF7KLsbW1ZdOmTfzf//0fSUlJbNiw4Yn+HT09\nPULVxvTT2tpaqN5ERkbS3d1NTk4OS5YseeJreNZ2XVVVFUlJSUydOnWQNX9fXx8XL14U3qOXlxdV\nVVWCqV5hYSGLFy/G2tqaBQsWsGDBAiQSCW+//TaHDh3C2tp62BH+scbNmzeZPHnykET54cOHdHV1\nCSJMkUjE9OnTycvLG5Hw3L17F4PBwOTJk6mtrUUmk41a7KxWq0lPT+fhw4fEx8fz2muvMXHiRMLC\nwpg9e7aQfWVa5MViMX5+ftjY2Aiak5ycHFQqFW5ubsjlco4ePcrixYuJiIggLi6O1NRUtm3bJkRQ\nFBUVcfnyZaGaYIpwkcvlzJgxQ8i7MhqNnD59mpMnT7Jx40YsLS2RyWT4+vqi0+mEsX6JRDKkSZ2j\noyMrV67k+PHjZro4kUiEn58ffn5+9PT0oFAoOHv2LC0tLTg4OHDmzBmampqwsrLCYDAgEomQyWSD\nSLzBYKCgoACVSkVRUdGofWOGg+l5VCqV4HVz9epVRCIR8+bNo7m5mUOHDrF48eIh21rl5eVmhEeh\nUDBx4sRB7s46nY7Tp0/z2muvDar0DtTwQL+OJykpieDgYEGsXFxcPGS+12jQ29vLrVu32LVrF9B/\nPEVHR7N161Zqamro7e3F0tKS3Nxcrly5QmpqquAz9k1pYQ2EVquls7NzWLuAbxpeEp4vGSUlJWRl\nZbFjx44R2zo2NjYsXryYlJQUdu3aZTaZYGdnx2uvvSb4pfz5z39m3rx5Qvrys8LLy4uSkhIqKyuf\n+TGeBNMis2TJEg4fPsyCBQuIiYlBJBJhMBh49OiRWfXGtNuWy+VMnTqVVatWDVqsFQrFqGIlngVG\no5Hs7GyuX7/O6tWrzYSZGo2G7Oxs8vLy8PX1xcPDg0WLFqFUKmltbSUtLY2YmBgcHR2FnffA78fL\ny4t169aRmJjIO++881yGaqOBTqfjxo0bbNmyZcj3mZGRQVxcnNmxFhoayoULF2htbR226ldUVISP\nj48QQjucePXx15KdnU12djbR0dF88MEHqFQq/vGPf2AwGFi9erWwqC5cuJCMjAwzh2h7e3szbZFW\nq6WxsRGlUonRaCQtLY20tDQcHBzo6uriT3/6k6CrMhqNvPLKK0RHR3P79m0CAgKGJPgikYgVK1Zw\n6tQpDh8+zMaNG7GwsGDGjBmkpKTg4+PDwoULUalUHDp0iB07dgwyjZw0aRLW1taUl5dTU1NDW1sb\nKpXK7KdarcbGxkYwXCwsLMTNzY2YmBimTJkiEB3T8Xb37l2kUikBAQH4+fnx5z//WdAjPS9MbS0P\nDw8qKiq4desW7733HmKxGHd3d2xsbKiurh40ceXv709WVpawoRhpFD0rKwuZTDYo0R7MW1rQvwkz\nGAw8ePBAIGS3b99+ZsJz69YtAgMDkclkwu+OHz9OQ0ODYPLY19dHU1MT//7v/87PfvYzoqKivrGh\nmq2trchksq80dPnLxEvC8yWirq6O06dPs3HjxlEx6ilTplBQUEBeXt6QScQymYw1a9YIido5OTnE\nx8cTFhb2TCeop6cnLS0tSCQS2traXgjrN+kSQkNDcXJyIjExkZs3b2JjY0NDQwMODg5C9Wbgbnsk\nqNXqUVcUnqbC09vby6lTp2hvb2fnzp3C56HT6bh58yaZmZn4+/uzc+dO6uvruXfvHtDfvvDw8CA9\nPZ3Q0NARy8V+fn4sW7aMhIQE3n33XbML8VjD1I4YqixfXFwMDA6qtLCwICoqihs3bgxZQevs7KSj\no4PXXnsN6K8Smf4/FEx6nwsXLuDp6cmuXbuQyWTU1NTw+eefo9PpEIvFZo6/06dPJyMjA4ClS5dy\n48YNampq8PT0FETKFhYWuLi40N3djaenJx0dHTQ1NQm+Lc3NzUgkEvR6PQ4ODohEIhoaGtDpdLS1\ntQ17XJhIT3JyMkeOHOGtt94SHJxnz54tHKutra0cOHCABQsW0NnZaUZo2tvbaW9vJykpCW9vb5yd\nnZHL5YSGhiKTyXBycuL06dM8fPiQ73//+/T19VFcXCzYU4SHhxMdHY2rqyspKSn8/Oc/Ry6X85//\n+Z9IpVKOHz9OaWkp+/btw93dnYULFz5z6LBMJqO1tZXOzk5OnDjBmjVrzDYYYWFh3L59G6lUilqt\nFtqxpmPcZGBXXl6ORCIZRIwePXpEXl4eu3fvHvL1DWxpmT7/4OBgHjx4gF6vJygoiFu3bj31+4L+\n8zYzM5P58+dTV1eHo6Mj9vb2VFRUDNIG9fT04OnpKUz+fVPxbRpJh5eE50uDSqXiyJEjrFy5ctRj\nyCKRiNdff51PP/2UkJCQYSMHPDw82LRpE5WVlVy8eJHr168/00XPJFx2dXWlqqpqTAmPXq+nsbFR\nCOr705/+JCxOXV1dtLe3s3Xr1meaQuvs7Bw2FPFZMTCH6I033hBSmhUKBVeuXGH8+PFs3rxZ0IaU\nlJQMqjKNVmsQEhJCV1cXBw8eZPv27S/EZds09fTqq68Ous1gMHDp0iVee+21IY+XmJgYPvnkE+Li\n4oSyfmdnJ62trVRVVSEWi3nllVfo7OxEpVIN26ZsaGjg3Llz9Pb2snLlSsEHp6mpif379ws7e6lU\nSnt7u0B4CgsLmTBhAnK5nJiYGLRaLbW1tWRmZpKeno6bmxttbW10dnYKJpGLFi1CrVZz6dIlIiMj\nSU9Px8PDg61bt9LW1kZDQwNKpZKmpiYePnzIrVu38PT0FDRBHh4ewhSfWCxmxYoVJCYmsn//fiIj\nI3F0dOTUqVM4OzujUqnQaDRIpVLOnj1LSEiI0PqysrIiMTGR7du3s2/fPubNmzekUL20tFSYsDKN\n95vGzAsKCti3bx9ubm6C9YFMJqOiokIY7d+0aRPt7e2cP39eiPSYP38+U6ZMeaqqr4uLC/X19Zw4\ncYLIyEizVl1LSwttbW1CBI6jo6NAeEQikdDWcnV1NUtF7+3txWg0Ym1tTUpKCrGxscOer4+3tKD/\n/Lh58yaWlpZMnTpVsIMY7SbHhMLCQkQiEenp6UilUrRarZDXZWNjYxZYamdnNyq37n92fJtG0uEl\n4flS0NPTQ0JCAnPnzn3q6RU3NzemTZvGuXPnWL9+/Yj3NUUU3Lt3j7S0NBwdHVm4cOGwBKu5uZk9\ne/Zw8OBBOjo68PDwYPr06cTGxlJVVfVchloDTf3q6upoaGhAJpPh4uKCtbU1b7/9Nm5ubmbl74SE\nBFatWvXUEwOmhW40GE2FxzSRYTIVMxqNFBcXc+nSJWxtbXnjjTcGpV4P5QP0NOLKadOm0dXVxaFD\nh9i2bduYu56Wl5cDDKk1KSwsxN7eflinamdnZyZMmEBRUZEQIllSUsKZM2cQiUQ4OjrS2tpKfX09\nfn5+gypynZ2dZGRkcP/+feLi4oiKijK7j4ODA9OnT+fmzZtCkGpHRweenp7CrnzFihXo9XoyMjK4\ne/cu7e3tuLi4IJPJaGhowN3dnQ0bNpiRCY1GIxjErVixggsXLgh5Vq6ursL0VGNjI3PnzqW6uprK\nykoyMzNpbW1Fq9ViaWkp5G6Z2k7Xrl0jPDycyspKpk2bhp+fH/b29hiNRo4cOYJOpxMcxh88eICH\nhwdubm4sXryYpKSkQa68SqWS3t5e5s2bN+izN/1dfHw89+/f5/Tp04hEIsaNG8fp06d588030Wg0\nTJgwAZFIRGhoqCCGvnDhAhcuXGDWrFnExMSMygrCxcWFrKwsnJychJwuE06cOEF9fT3Qf2w/viHy\n8/Pj3r17BAQEmI2iX758mfz8fGEQYySfr8dbWtDf+tXpdDg6OiKTyQRz1sf9gEaCwWAgKyuLgIAA\niouL0Wg0QL90YOfOnVy7do1Lly4J3/e6deuG3Bx809DS0vKl6Qe/DnhJeF4wdDodR48eJSAg4JmC\nOQHmzZvH3/72NyEvaCSIRCJCQkIICgqisLCQL774Ah8fH+Lj482YvCl7x2g0CgZcdXV1FBcXc+jQ\nIb73ve+xcuVK4f4Gg4Hz58/j7u4+qMyr0+loaGgw095otVqh3B8bG4tcLsfKyoqamhrOnz9vZtgl\nEomYMWMGnp6eJCYmEh0dTWxs7KirU08yHRwtdDodaWlpVFRUCDlE5eXlpKenYzAYePXVVwkMDBzy\ndanV6kHVqaedJomNjaWrq4sjR46wadOmMfXoycnJGXK8XKfTceXKFdatWzfi5z19+nTOnTtHdHS0\nQHIsLS3p6+ujvb2dv/71rwQGBpq1xHQ6Hbm5uWRlZREREcEHH3ww5KJrY2PDq6++SnFxsUA8NBoN\nN27cQKFQ0NPTQ2JiolC9MVWGTJNvJm3SgQMHCA4OJi4ujtbWVk6cOIG/vz+TJ0/m6tWrxMTEkJyc\nTGxsrNByqq6upqWlhaKiIkEjIpPJmDhxIra2tsL50d7ejlKpRK1WYzQauXPnDhMmTODevXuEhIQI\nye/r1q1j//79XL16ldjYWJRKpaDNioiIoLy8nLS0NOG1Q78wWCaTjWiqJ5FICA4OJjk5GT8/P7y8\nvKioqODAgQN4e3sLZp4SiYSYmBgiIiLIy8sjMzOT/Px8MjMziYyMFBzdh0N3dzft7e28++67g4jr\nm2++yd69e4WQ18dN9/z9/Tl79ix5eXlmo+gm8lhUVIS7uzttbW3D6sGGIjytra1mBNHT01M4VkaD\npqYmwWepqalJ+L2trS3bt2/n4sWLzJ07lx/96Ee0tLQQHh7+wo1evy5oaWkRNjHfBrwkPC8QpikP\na2vr59otSKVSli1bRkpKCu+///6oRk8lEglTp05lypQp5Obm8o9//IPQ0FBiY2MpKipiw4YNZk6j\nJvT09NDT08Mf/vAHtm/fTmBgIN3d3SQkJNDQ0ICnpyf+/v5m6c5NTU3ChExQUBALFy4c1tRvJNPB\nCRMmsHv3bo4dO0ZdXR1vvPHGqISYTxMrMVyFp62tjWPHjuHo6MiuXbtoaWnhwIEDtLe3ExcXR2ho\n6IiEoKOj47kqPNBP/JYuXUpSUhLHjx9n3bp1YyImbGpqQqlUDunRcvPmTTw8PPDx8RnxMfz8/NDr\n9YJg1cnJSdBamPQseXl5LF++HKPRSGlpKRcuXMDNze2JwahdXV1CeOSdO3dQKpUolUq8vLxoaWlh\nxYoVhIaGCp9Fbm6uYEUA/efHrFmziIiI4MKFC/zhD38A+gXDOp2OkpISurq6uHr1KhKJhIyMDLy9\nvZHJZAKB3bRp06iONY1GQ319PWlpaTQ2NtLZ2cmHH36Ii4uL0AqbP38+aWlpODk5oVQqhaquqUX9\n97//nbt37wqL6sOHD4VgzpHQ2tqK0WjEx8eH2NhYoqOj+d3vfodWq+UPf/gDQUFBREVFCRqjOXPm\nMHXqVK5fv86NGzeorq5GoVAQGBjIrFmzBhH0rq4uzp07N6wA2sHBgd27d7Nnzx7a29sHESd7e3vs\n7e0pLCzku9/9rvB7k8M49Gt4UlNTeeedd4Z8j49reKC/Fers7CwQrbCwMNLS0kas1nZ2dnLnzh0U\nCgVdXV3o9XpeffVVpk+fztGjRykvL+eNN97g6NGjdHR0sGzZsjFxmP9ngslt/aWG5yXGBJcvX6al\npYWtW7c+98IVEBCAt7c3V69eZeHChaP+O0tLS+bNm2c2yv75558PSXYGQqvV8uGHH/Kf//mfHDp0\nSDA7q6urY+/evfj4+ODt7U1ISAheXl6j9v94UnCovb09W7Zs4eLFi3zyySds2LDhiSGQnZ2dz1Xh\nKSsr4+TJk8yePZvAwECSk5Opq6sjNjaWyMjIUZEWtVo9SGP1LH4hJv+WhIQEUlNTef311597QiQn\nJ4eYmJhBFSONRkNmZiabN29+4mMMHFEfSHhEIhHx8fH4+PgIrYKDBw/S2dnJsmXLBrXJTFqugdXA\n7u5uLC0t8fDwELKmbGxsuHnzpuCtZEJvby9qtZrW1laysrLMxMEm8bFUKsXa2pqHDx8SEhJCXFwc\nLi4u3Lp1izt37tDV1cXixYuxsrKivr6e+/fvj3rCycrKCj8/P4GYNzQ0EBUVxaRJk2hoaKChoUF4\njlOnTiGRSHB0dMTOzg4PDw/s7OxYu3YtCQkJyOVy2tra6OvrGxXhqaysxNraWlig2trasLe3x2Aw\n8MEHH3Dnzh1SU1PR6/VERUURGRmJvb098fHxTJ8+nWvXrqFSqYQqoqurq3DMQ3/LKiIigpKSEtra\n2oaMTbCzs2P37t189NFHqFQqDh8+TF5eHg0NDUgkEjo7O4UWlAkmwmMipiONzw+l4WloaMDV1RWV\nSoVKpWLKlCmcOXOGxsZGs8lGrVZLaWkpt2/fprq6muDgYBYvXoxWqyUjI4Np06YB/U7jxcXFHD9+\nXMg1+zZVOUzo6upCLBaPWVzHPwNeEp4XhIKCAm7fvs2OHTvGLEWBxH+EAAAgAElEQVT31Vdf5eOP\nPyY8PPypM1xsbW1ZsmQJXl5e/OIXv3ji/XU6HYcPH2bixIlmFy+JRMLWrVufOUNmNEnpEomEJUuW\nIJfL+fzzz1m8ePGwlvl9fX0YjcZRe2QYjUaBfBqNRq5cuUJ+fj5Lly6lrKyMrKwsZs+ezRtvvDHq\n781oNA5qqxmNxmc2SJNKpbz55pt89tlnXL58mbi4uKd+DBO6urq4d+8eH3zwwaDbcnJy8Pf3H7Xb\nc0REBJcuXaK1tVX4DqdPn87MmTO5cOECYrGYzz77TLBIkEqldHZ2UltbS01NDXV1ddTX1+Ps7Iy3\ntzcTJ05k3rx5ODk58fvf/57Vq1djY2NDe3s71dXVpKen88orr3D06FGB2BgMBqysrJBIJIwbN45x\n48YRGBiIUqkkNzeX2NhYQSxbV1fH+fPnSUtLY9GiRSxYsACtVotCoRBE2qaE9aeFRCJh3bp1HDhw\ngOvXrzN37lyzionBYCA/P58zZ87Q1dXFtWvXUCqVWFlZCVN8n3/+Oba2tri5uY3qWDOZL5oIT1FR\nETExMVRUVFBcXMzMmTOZMWMGdXV15Ofn89FHH+Hr60t0dDSBgYEsXbqUWbNmcfnyZZqamrC3t+fi\nxYtCJU6j0RAfH09TUxOtra1DnuNGo5Fz587xxRdfUFxcLHzHJrsAkUiElZUV+/btY9myZfziF7+g\nr68PT09P1q9f/8QpxKFaWkqlUtD/3bt3j9mzZ2NnZ8etW7dYtmwZVVVVKBQKSkpK8PLyIiIignXr\n1gnXhP379zNnzhxEIhE6nY7Lly/z8OFDli5dytmzZ3nvvfe+sWPnI+HbJliGl4TnhcCk+9i2bduY\nTtw4ODgQGxtLSkqKYKT2tHj06BG2trZmZebh0Nvby/vvv49araa8vJz79+8LdvovkvCYEBYWhru7\nO0ePHqW2tpbXXnttUJXCZHU/3GeRmZnJr371K27cuIGNjQ2xsbGsW7eO7u5uTpw4QU9PD4GBgZw5\nc4aYmBj+5V/+5alzvrq7u7GwsDBbtEzTH896IbWysmLTpk18+umn2NnZMX369Gd6nBs3bhASEjJo\noqW7u5vc3Fx27tw56sfS6XSkp6fzH//xH+j1ejw8PFiyZAk5OTnk5OQQGBhIeHg4ly5dQqFQ0Nvb\nS29vL97e3sjlcubNmyfs/k1VmeLiYioqKgDYu3cvXV1dODo6IpFIsLS0xN3dHWdnZ0FbY2NjQ35+\nPvX19SxZsoTu7m5Onz6NSqVi27ZtZiP3crmcbdu2UVpayrlz53BycmLRokV0dXVx48YNIiIicHFx\neSbCA/3EdMuWLfz2t7/lwIEDbNu2TSDTJt8akznijh07cHBwQKVSoVQqhWpQS0sLFhYWHDx4UJgQ\n8/T0xMXFxezYMRqNVFZW0t3djaurKwaDgeLiYrZv305ISAj79+8nODgYR0dHQTu3ZMkS7t69y7Vr\n10hJSSEiIoKoqChWr15NU1MTly5dEr6f0tJSrKysyMrKEkToj6Ouro5NmzZx8+ZNs4mmgTAajYKf\nzcmTJ0lLS2PLli38+Mc/HtV59TjhMRqNKJVK3N3d8fDwEAiPh4cHd+7c4cGDB1hZWREeHs53v/vd\nQVXWmpoa2tvbCQsLo7W1lcTERJydndmxYweHDh1i0aJFw06/ftPxbWtnwUvCM+ZoamoiKSmJ9evX\nv5CDKSYmBoVC8cQk6OFgmkIYDUy2+RMmTGDChAksWLBAWMifFT09PU/VfnJ3d2fXrl2cPHmS/fv3\ns379erOR1pH0O3/+85/5f//v/5l5bHzxxRecPn2a73//+0yYMIHW1lbkcjnvv//+M5PTsWpnPQ47\nOzveeecd9u3bh62trRBhMVqY/IK2bt066LbMzExCQkKeKkJk7dq1XL16VQhwrK+vJzY2lp/85CdI\nJBLKysq4f/8+0H/szJo1S/C5qa+v5+7du7S1tWFlZYVMJhOITHd3NzExMUydOhVHR0eMRiN//vOf\nWbdu3ZDaIpNJ5cOHD0lOTiYsLIy1a9cOKfI2+bi88sor5Ofnc+jQIQICAnBzc+PgwYP867/+Kzqd\n7rmqccuWLSM1NZWTJ0+yevVq4fxQKpX4+/vj7Ows+Cy5uLjg4uIifPanTp3i9ddfF3yo7t69y8WL\nF4XYDBMJsrW1RafTYW9vj4WFBWVlZTg7OwvfX0xMDGlpaWzYsEF4bVZWVkRHRxMdHU1TUxMFBQX8\n4x//YPz48URFRbF27VoqKys5cuQItra2zJw5k/r6ekpLS3FxcSE4OFioyBQWFhIXF0dnZ6dZxXck\nmATfpirYlStXRlXhGajhMQXs6vV63NzcKC4u5uOPP6a9vZ3e3l7eeecdvLy8RtzwzJ49W5gqjI2N\nZdq0aWRmZmJra/tcYav/7HhZ4XmJ54JarRbSfV/UqJ9YLOb1118nISGBSZMmPXX/NTIyctSJw15e\nXjQ1NZmdFM+rRert7X3q6pCVlRUbNmwgKyuLvXv3smbNGmG8ergJrYqKCn784x8Lu00TtFotHR0d\nHDp0iN/+9resW7fuuf2GxkKwPBxkMhmbNm3i4MGD2NjYDDs6PhRMRoOPf94dHR0UFBSYCUufhAcP\nHnD58mVhnNcEvV5Pfn4+06ZNM9uZt7a28vDhQ6EVERAQIJCcge1HtVrNjRs3mD9/vlAhu3XrFm5u\nbsMKqY1GI7W1tZSXl7N69eohR+0fh0QiYdq0aYSHh5OVlcWDBw/o6+tj//79WFtb09vb+9S+LiaE\nhoZy8eJFWlpaSE5OZtWqVYjFYhoaGgSH8Pb2do4ePcrGjRuF46KgoABXV1fS09N57733mDRpkvCY\nPT09QiWovLyciooKIUfq9OnTNDY24uPjg1arxcLCQpjkLC0tHdL6wt3dnSVLlrBw4UJKS0spKCjg\n7NmzWFlZERISQkREBKmpqbS2thIWFkZtbS179uzB398fDw8PXn/9dUE0/LTo6emhtLSU2NhY8vLy\nRqz0PK7hqa2txd7entLSUnp7e7G3t8fb25udO3fyq1/9CrVaPSzZaWpqoq6uTgi03bRpk3BNy8nJ\nGdb88NuClpaWJw4rfNPwkvCMEfr6+jh8+DDR0dGEh4e/0Ofy8vISLrIDR8dHA3t7ezZv3sy+fftG\nJD4WFha8/fbbVFZWDnLffR48TUtrIEQikaCTOHHiBDNmzGDOnDnDVng++eSTYQNKTeLr6dOnj4m5\n4vN68DwJ48ePZ/369cKCORrPIVMkxlCux1euXCE6OnrEUr5arRa0N1VVVWRnZw+5OJhEyOPHj0et\nVqPVatHpdFhaWvL2228/8XUWFRURHBwskB29Xs+1a9dYu3btkPdvbGzkypUriMVivvOd7zx1nIKV\nlRXx8fHCiLrJEfhp3Lofh1gsZvr06SiVSjo6Ojh9+jQrV65EqVQKoaJLly7liy++ICUlhZUrVwrH\n4OrVq2lsbCQ5OZm3335b+IxtbGyEvC3obw/19vYiFotxc3NDoVCg1Wq5efMmMpkMT09P/Pz8OH36\nNJ6ensN+t1KplNDQUEJDQ0lPT0ehUFBVVUVLS4tQIbtz5w4SiYSNGzdSX1/P+vXrUavVz/TZmNDX\n18fDhw/56U9/yu9///sRP0vTNKBCoeD27dvY2dlhZWUlxFTk5OQglUqRyWQUFhYOG7576dIlxGIx\nXV1dvPfee1hbW2MwGEhOTiY+Pn7MzUr/2fCywvMSzwSDwUBiYiLjx48f0jzsRSA+Pp6PPvqI6urq\nQSZ4T8KvfvUrzp49i1KpHJL02NjY4OXlhZOTE0VFRcydO3fMsqp6e3ufK/PHFOVgGl032cM/jpKS\nErPE8sdhbW1NdXX1Mzk7P46Ojo4X0tIaCF9fX1auXMmRI0fYunXrE9ul5eXlQkjlQLS2tg4SMev1\nesFHqbq6murqavr6+rC0tESn02E0GgkMDByylWFlZcW4cePYsGEDMpkMpVKJQqHg0aNHo3pft2/f\nNiNlCoUCV1fXQTtPo9FIbm4u165dIzg4GK1W+1zHkaOjI5s3b+bgwYOUl5fz6aefsmbNGoKDg59p\n1x8dHc2f/vQndu3axalTp0hOTqa5uVkQhIvFYtauXSt49Dx69IiOjg5CQ0MJCQlh37595OTkmOWD\nDURVVRVeXl74+vpib2+Pr68vmzdvRq/XC7YDDQ0NGI1G/vjHP+Lk5GTmGu3p6Wl2ntTW1lJQUMDO\nnTtxdHREoVBw6tQp4Xa9Xk9SUhIKhYKOjo5Rt8FHQnd3N3//+9/ZvHnzkJENra2tFBYWYjAYOH36\nNOHh4fj4+BATE8PVq1dxcnLCzc2NkydP0tXVRWBgIHfu3BnyuW7evElJSQnx8fHMnTtX+E6vX7+O\ntbX1Nz4y4knQ6/Uj+iF9U/GS8DwnjEYjZ8+eRa/Xs3z58i+tRGplZcWSJUtISUnhvffee6rF1dXV\nlZs3b7Jp0yYyMzOB/hFlW1tb9Ho9mzZtIj4+noCAAM6dO2eWyv4s1ZmBeNYKz0A4OTmxbds20tLS\nKCwsZM6cOYPu4+vri1QqHVZvoNFonjjuPlqMhengaBAUFER3dzeHDh3i3XffHbFCk52dLbj9DsSl\nS5eIioqisrKSmpoaKisrefTokdBm6uvrw9XVlcmTJ+Pj44PBYOD69etAv06ksLBQaBNKpVIsLS2J\njo7GxsYGkUgkiG5HA5PDsClvyVTdWbNmjdn91Go1J0+epK+vjx07dlBbW0tZWdnoPrQnYPXq1fzh\nD3/A0tKS06dPk52dzauvvjpsPMZwsLGxITQ0FIVCwcaNG/n000+RSqVmuiJLS0s2btzIn/70J37+\n85/j4uLCr3/9a6BfH7V37158fX0HHUvt7e1otVq6urpwc3Pjxo0bgp5LIpEIn3lUVBQLFizgr3/9\nK4sXLxYMQbOzs2loaEAqleLp6SlUiExVDpMjNCAYKJocr0+fPj0oZ+p50Nvbyy9/+UuSkpKA/nbX\n3bt3USgUqFQqwZvou9/9LmKxmBs3buDp6Slkq1lYWBAYGEhJSQlTp04lLy+Pnp4egfzq9XouXLhA\nQUEB4eHhZhvQR48ekZ2dza5du77VrSzojzpydHQcU3PTfwZ8u97tC0B2djbV1dW8++67Y77APQkh\nISEUFhaSk5Mz5KI/Etzd3blw4QKVlZUkJyfT0dGBhYUFPj4+AhFqbW1l4sSJhIaGUlNTw1/+8hfm\nzJnDtGnTnvlEGQvCA/2L7fLly6murub69eu4uLiYiXp3797NJ598MiThEYlEhIWFDQo2fFa8KNHy\nUIiKijKLoBiqytHU1ERjYyNvvfUWOp0OpVJJTU0NJSUlVFdXI5VKsbCwEOISgoKChKwqDw8PpFIp\nLS0tXLhwgcbGRjw9Pamrq+Po0aMkJyfz3//930gkEhYvXkxISAjOzs588sknrF+//qkqZgqFgvDw\ncGHxUSgUuLi4mFUs7927J0zQzZ8/H7FYTF1d3bDtyqeFg4MD48ePR6vVCoZ7R48eZcKECYKB5mgx\nY8YMYSx/6tSpXLp0idTUVJYtWya8R5PP1JEjR8yOP5lMJphO7t692yxapLKyEl9fX6qrq4Wwy1Wr\nVg35GmxtbVm0aBHXrl1j586dgoeR0Wikvb2dhoYG0tPTsbS05MqVK1y8eFEIDYb+c8PBwQFra2vB\n8mEsYTQaOXPmDDk5OVRXV1NeXk5gYCDz5s0jICAAiUQiRIz09PSg1Wpxdnamu7tbONYnT55MQUEB\n0dHRSKVSFAoFM2fOpK2tjcTERKytrRGLxSxatEh4XoPBwKlTp1iwYMELCUX+Z8O3sZ0FLwnPc6G4\nuJicnBx27NgxJov400IkErFs2TL27NlDaGjoM53IEydO5Ac/+AEA58+fJzs7m8zMTLy8vLhy5Qr+\n/v40NzezatUqGhsbycjIIDc3l7i4OKZMmfLUIuaRnJafBaY063PnzqFQKHBzcxOqFu+++y779+83\n26GaggL3798/Zq/hRYqWh8KcOXPo6uri8OHDbN682Wwcvr29nbNnz+Lk5MQnn3wijD2bduwymYwp\nU6YIo+KPi957e3vJyMigsLCQadOmodVq6e7uZvfu3djZ2bF48WIcHBzYunUrNTU1fPrpp7S3tzNz\n5kwOHTpEXFycoFsZCQaDgTt37ggTZI9Xd/r6+jh79ixVVVW89dZbZhUXsVg8pguxj48Pd+/eZdGi\nRWRlZRETE4PRaGTPnj1EREQwf/78UbXPxo0bJ4xLt7S0MHPmTO7fv09aWppZMGtjYyM/+MEPhCiX\nkJAQoN+G4eHDh6SmpppVuUztrAcPHlBbW4u/v/+I51BERAQKhYK8vDxmzpwJ9J8nzs7OlJSUYGVl\nxfbt2wWjQOX/x96bR0V1r+n+nxqoogoKGWVGGQUFGUVAIzgiihoS5ylGY5KT4zn3rNvd667uXt33\n9Hj6nu6+tzsxJ4MmakxMFDVoUBBREGQSRUAEFAGZB5nnoqDq94e/2scSUDDmnCT6rFVr6S6q2FXs\nvb/Pft/nfZ7mZk6fPg08/Lvo090LCwvp6+t75u91IohEIo4fP86OHTtYu3btmM+iFy7rYzn0bVX9\nce7p6cnZs2cZGhrC1taW0tJSzM3NSUxMZMGCBQwMDGBubm5wTup1Py+am/JEeFEJz/f3rX9BUVdX\nx7lz59iyZcufVfxmYWFBeHg4SUlJ33sR6OzsBBCMyZqbm3FxcaGmpgZ4KJ7dsmULcXFxXL9+nU8/\n/ZSKioopjbkPDw8/V8LT29uLvb09fX19VFZWkpubS3NzMzKZjA8++IDDhw8LidEKhYLFixfzF3/x\nF1y6dIm6urrntg9/qgoPPFwwVqxYwbRp0zh69ChXr17l6NGj/J//83/44IMPuH//vqDnCAwMZNWq\nVaxZswaVSsV7773H4sWL8fT0NCA7Wq2WGzdu8OGHHzI4OMj69euF7KMdO3YIgt47d+4I00R6EqBW\nq8nKysLJyYm8vDyh/fQkVFZWCpoMeKjlsbCwwMXFhfr6ej755BMA3nnnnTHtJX3g7POCUqnE1dWV\n9PR0Nm/eTGFhIQqFgvfeew+NRsP+/fvJzs6e1Dh2WFgYubm5NDU14eTkxPbt26mvryclJQWdTodO\np+P69etERUXh7e1NQkKCwfRTTEwMjY2NFBUVCdtqamowMzPDysqK27dvP9WeQB9hkZGRYeC31dDQ\nQGZmJuvXrxeOTVNTU1xdXRkaGsLIyAiZTIaHhwdubm4UFRU99woPPJyUNDMzIygoaNxrgd6Lp6mp\nCTs7O6FlpSeMMpkMNzc37ty5g7e3Nw0NDSQnJ7N582aCgoK4ceOGgXN1e3s7V69eZe3atS98K0uP\ntra2F5LwvKzwPAM6Ojo4ceIEr7766nPTgXwfRERE8PHHHwsXgGdFc3Mz8HCxTk1NxdjYGIVCwYMH\nD1Cr1UKZfcaMGezevZs7d+6QkpJCVlYWy5Yte6ruQa1WI5PJnttFZ3R0FLVajYWFBZGRkVy5ckV4\nbv78+YjFYjZs2CCkzGdkZHDlyhWcnJyora3lyJEj2NrasmzZMmbOnPlM+6XRaBgeHh5TKXnehEen\n09HT00NdXR3V1dXcv39fiFKoq6vDxMQEFxcXRkdHkUqlrF+/Xmg76nQ6jhw5QlRU1LityPv373Ph\nwgVhsqq9vZ1Tp06xcuVKg1gHnU5HRUWFENrY0tIiPDcyMsLdu3eJjY2lrq6OgwcPsmHDhgktCIqL\ni/H39xe+q4yMDNatW8eVK1fIz89n1apVQuXjcTxvwqNQKDA2NsbR0ZHi4mJ27NjB4cOHkclkxMbG\nMn/+fFJTU8nPz2fJkiX4+vpOeKy4u7uTlJRER0cHdnZ2GBsbs337do4ePcrFixfx9vZmeHgYNzc3\npk+fzvvvv8/Ro0eFKrFMJuP111/n6NGjODk5IZPJGBwcRKvVMm3aNKHi9TRYW1sTGhpKUlISmzdv\nZmhoiFOnTrF69eoxbTq91YVeCyQSiSgvLzfIK3ueGB0dpbKycsLn9YSnubkZT09PQb/zKHx8fCgs\nLBS+m9dffx0nJyeys7Nxd3cXxLj6qazIyMgptSd/7mhvbzc4t18UvKzwTBF60eiiRYuemlz+p4JE\nImH16tUkJSWN8UmZLEZHR+nu7hbcgeVyOSqVitbWVhwcHMZUQ/SGbr/4xS+YO3cuJ06c4MSJE7S1\ntU34O77vhNbj6Ovrw8TEhNbWVm7duoWJiQkikQgTExPy8vLIy8szWBi7urrQarW0trYKZmaNjY18\n8cUXVFVVPdM+TOT0/H0Jz8jICLW1tWRmZnL48GGhenPmzBmKi4tRKBRERESwYcMG7O3t8ff3Z8OG\nDTQ3N7N06VIDYlNZWUl/f79AMPTo7OzkxIkTJCQksGDBAnbs2EFxcTFpaWns2LFjzAWxsbERY2Nj\nYTGprq5GJBIhlUoRiURs2rSJ4OBg1q1bR1hYGIcPH+bWrVtjPtvQ0BAVFRWCQLW4uBhTU1MuX75M\nTU0Nb7/99oRkB34YwjM4OEh0dDT5+fnodDp27NjBpUuXuH37NjY2NmzZsoV169aRk5PDwYMHharn\nePs2d+5cdDqdsEgrFAp27NhBdXU1iYmJQstPpVLh5eWFQqEgPj5eMNyzs7MjMjKSU6dOUV1djYuL\nC+3t7YyMjDBr1qxJR54sXLiQtrY2ysrKOHv2LB4eHuN+ryKRiODgYAMDP0tLy+emkxoPT6qW6UfT\nm5ubsbe3N9DvPPoz1dXV+Pj4oFAoKCsrY2RkZIye8dq1a0IO3Ev8ES9qS+tlhWcKGBkZ4fjx43h7\newtBdD8WzJw5UyjLR0dHT/n1Wq0WNzc3XFxcyMzM5L333qOwsJDGxkZmzJjB/fv3hZDBRyEWiwkK\nChJS2Q8dOoS3tzdRUVFjdC3PS7CsR09PD2KxmC+++IIVK1bg6enJxx9/TExMDLa2tpw4cYKGhgZi\nY2ORyWQ0NTUBCEGo8PBiHxUVNSnzuvEwXjsLpkZ49ILS2tpa7t27R11dnfDZ9Hf2s2bNws3NDWdn\n5zFJ9DNmzODQoUN0dXXh4OBgUFXR6XRcunSJxYsXC3orfWjojRs3mD9/PnFxcajVar788kuMjIzY\nu3fvuMT07t27QjtLp9NRWVmJo6MjPj4+tLe3U11dLYx1BwUF4eDgwIkTJ6itrSU6OlogYaWlpbi6\nugruwampqYyOjhIZGUlYWNhTK20/REtrcHAQMzMzIiIiuHDhAlu2bBEqM0ZGRnh5eTFz5kz27t1L\nSUkJ3377LXZ2dixbtmyMRYCFhQVarZbOzk6hqqBQKHjttdf46KOPhOwp/UL83XffYWlpKXj0iEQi\n5s2bR1VVFVlZWQQEBFBfX09HR4egyZkM9ML+48ePY2ZmxmuvvTbp11pYWCCXy38wDc+TKuMSiYSh\noSF6enqwtramra1NII+jo6MCEXVycsLS0hInJyfu3r2LlZUVtra2QqBoR0cHGRkZvPXWWy9bWY9g\naGgIjUbz3KxGfkp4WeGZJHQ6HWfOnMHU1NRA/f9jwvLlyykuLhZaU1OBkZER27dvZ9GiRbi5uVFV\nVYW9vT1NTU3ChMjTXr9w4UL27duHsbExH330EZcuXTJwOn6egmW1Wi1Y8O/atQt/f3+USiW/+c1v\n8PHxwdLSkj179iASifjss89oa2sTSvT6xVIikeDo6MjAwMAzXxDHEyzDkwmPRqOhtraWy5cvc/Dg\nQX73u9/xwQcfkJCQwL1797C0tGTp0qXs2rWLv/7rv+bXv/41cXFxQvbT4/uqVCrZtm0b5eXlBllS\n8JBciEQifHx80Ol0FBYW8uGHH9LT08O7775LZGQkra2tHDhwABcXF7Zs2TJhFe5RwtPe3o5Wq2X3\n7t1ERESwdOlSSkpKDNpcdnZ2vP322/T19QmEDP7YzhocHBQMMHft2jXuGP14+KEqPADh4eG0t7dz\n9+5dQbOmNyjU/24/Pz/27duHs7Mzhw4dEgJC9Whra8PBwYFr164Z/J6ysjLmzJlDZWWl0H51cXFB\nIpEQEhJCa2ursF0kErF27Vra2toQi8W0tLQwODg4ZWIul8vRaDTC9N1kYWRk9NymGB+HTCZDq9VS\nX18/7vNisZjW1lZsbGwQi8VChae7u5sjR47Q1tbGO++8Q2BgIGVlZfj7+9PR0SGEuMLDc/zs2bMs\nWrTohfOaeRr0+p0XkQS+rPBMEpcvX6arq4udO3f+aA8UExMTli5dSmJiorDYPwu8vLyoqKggJiaG\n5uZmHBwcBJPCp5XTFQoFy5cvJzQ0lPT0dD744AMWLlzIvHnznluFp7m5mZMnT6JUKpk9e7ZBRePR\nqTEjIyNeffVVrl+/zueffy6MHutL+IGBgZibm3PgwAGcnJymnFUFE0db6AmPTqejq6uL+/fvU1FR\nQUNDg2CHLxKJsLa2JigoCHd393GnpiaLtrY2LCwsuHnzJi4uLnh5eaHVaklLSyMmJob6+nqSk5MR\niURs3LhR0FsVFBRw6dIl1qxZ80T9l35yR28IeO/ePdzd3YVjTKlUsnjxYpKSknjjjTeE7cbGxmzc\nuJHc3FwOHjzIkiVLePDgAVKplI8++ojh4WE2btwo3JVPBj8k4ZFIJMTExHDu3Dnc3NxwdHRk48aN\nnDhxgk2bNgkj81KplAULFhAYGEhGRgYffvghYWFhhIeH09zcjL+/P5cvXyYqKgq5XC6Iwrds2YJK\npeLIkSOIRCIh2+nmzZts2bKFzz//HHNzcwICAtDpdEgkEjIyMhgaGiIgIGBKU5FqtZqTJ08SExND\nWloaDQ0Nk3Lp1iMgIIDy8vLn6sMDf6yqnjp1CpVKRUREBF5eXgbBqy0tLcIxMTg4yODgIAcOHCAs\nLExIPvf29iYlJYXY2Fh0Oh1isVj4++Tn56PVal+2ssbBi9rOgpeEZ1K4ceMGpaWl7NmzZ9L98z8X\nAgMDKSws5MaNG888gunp6cmlS5eQy+WYmJjQ29uLra0tdSCfPPsAACAASURBVHV1k77DnDZtGuvW\nraO1tVUYZXd1dTXwF5kqdDodN27cIC0tjejoaNrb2yfV/pg3bx62trZCab+trY2IiAhhjH/Dhg18\n+eWX2NraTjnn63GXZY1GQ319PQUFBTQ0NPCv//qvaLVadDodKpUKR0dHPD09mTFjxpjW1PdBbm4u\nCxcuxNramq+//prNmzfz4MEDFAoFhYWF1NTUsGzZMvz8/BCJRIyMjJCUlCR4SD3Nufnu3bt4enoK\ni1JlZSWBgYEGP6OfkCkpKTHQ/4hEIsLDw3FycuKrr75CKpVy5swZ/Pz8aGpqGrdV+iTojfGeFx4l\nPPBQeGxnZ0dWVhaRkZHMmDGDuLg4jh8/LuQx6aFUKlm5ciWhoaFcunSJDz74AI1Gw8qVK6mqqqKo\nqIjQ0FDu3LkjuB8D7Ny5kyNHjiAWi5k/fz6XL19mdHSUrVu3cvjwYVQqFWq1mpkzZ2Jubi6ku08W\nOp2OxMREZs6cSXBwMFKplMTERPbu3Ttp0rRs2TK+/PLLSf/OycLX15e4uDi0Wi1lZWVkZmZy8eJF\nwsPD8ff3RyKR8ODBA9zd3dFqtYKAeuvWrQYeTUqlEgcHB+7fv49YLEapVCISiejs7CQ9PZ3du3d/\n7+y/nyNeZMIj+e1vf/vbP/dO/Jhx7949kpOT2blz508ie0UkEuHg4MCZM2fw9/c3CGqcLORyObdu\n3cLW1paenh5kMhlGRkb09PSMiSp4GkxMTPD19cXBwYGcnByhVD1ea+ZJUKvVQstn27ZtuLq6UlJS\ngpWV1aTM7qZNm0ZAQAA5OTlotVpMTU2FaoVKpUKpVJKcnIy/v/+kS/86nY5r164xPDxMQUEBKSkp\npKWlUVxcTE9PD1KplLCwMJYuXUpMTAwLFixgzpw52NvbG4zZfl+0traSnZ3N2rVrMTc3x9bWlvj4\neO7evcvg4CCenp6sX79eEKXqw1P1eUlPcmzWIz09nTlz5jB9+nQ0Gg1JSUmsWrXK4LsSiUTY2try\n3XffCaZwj0KtVpOfnw88FMU2NDSwcuXKKftHdXV1UVtbOyUC8CRIJBLS0tJYuHChsEA6Ojpy9uxZ\n5syZIwi1raysOHXqFB4eHmNyt/ROy1ZWVhQVFVFdXY2Xlxf5+fnCtFRwcLAQNSGTyfDx8eHChQto\ntVrMzc1pbW1lzpw5ODk5cfLkSUZHR3FycsLU1JR79+5hYWEx6TZTQUEBFRUVbNy4EYlEIvjVDA4O\nPjUwsrm5maysLMrKyuju7qa1tfW5EUyZTMaOHTsIDQ3F2NgYGxsb+vr6CAwMpKSkhIsXL6LRaFCr\n1cyePZvExES6u7t55ZVXBJH7oxgeHhamtYaGhggPD+fEiRMEBgaOG6L6Eg9jN1xcXIRj8UXCS8Lz\nBDQ3N3P8+HE2bdr0oxg/nyxMTU0ZGBigvLz8mYM/u7u7aW9vx8LCgvb2dlxdXbl16xYBAQHP9H7T\npk2jr68PpVJJcXExpaWlWFtbT2qxbWpq4ujRo9jY2LBhwwahhVRQUMDMmTOfWp3QQyaT0dHRwdDQ\nEGVlZXh4eAjvZW9vT3NzM2VlZfj4+IxLRoaHh6muriYrK4vU1FRSUlJob28XcqNcXV2JjIxk1apV\nTJs2DZlMRnR0NGZmZj/onWZqairu7u64urqi0+lobGykvLyckZERduzYQUBAgKAnun//Pl9++SX+\n/v7ExMRMitwNDw+TnJwsEJzq6mo6OjrGFe5PmzaNlpYWmpqahGR3nU5Hfn4+p0+fRi6X8z//5//k\n3r17NDc3M3fu3GciPDU1Nc98LD4OkUhEXl4eQUFBwg2CsbExo6OjFBYWCq1Oa2trVCoVCQkJeHl5\njdt+7Orqoru7m4ULF5KVlUVfXx8ajYZ79+6xZs0ag+NALpfj7e1NcnIytra23Lx5k9DQUCFVPiMj\ng9DQUAoLC5HJZNy/fx9nZ+en3ni1tLSQkJDAtm3bhONbJBLh5OREQkKCQOIeRW9vLzdu3CAxMZGi\noiIcHBzw8vLCxcWF9PR0Ay3es0IqlRIeHs727dtJTExkcHCQ7u5uUlJSGBkZYfPmzXh5eVFQUEBv\nby/l5eXMmjULiUSCq6vruOe5mZkZFy5cIDQ0lMrKSuRyOS0tLaxZs+ZHKz34cyMjI4OgoKCXouWX\n+CN6enr4+uuviYmJmXI4548BkZGR1NTUUF1d/Uyv1+t4HBwcaGpqwsXFhcbGxkmZr00EtVqNs7Mz\nv/jFL/D39yc+Pp7jx49POMquXyi//PJLFi9eTGxsrEFLUT8SPlXo2y1fffUVBQUFwvaYmBja29u5\ndu0aOp2OtrY2rl69ypEjR/j973/P7373O44dOya0J5YvX46JiQnvvfce+/btY926dXh7e6NQKH5Q\n48FH0dfXR3l5OSEhITQ0NPD555+TlZWFRCIhPDycs2fPMjAwgE6nIzc3l5MnT/Lqq6/yyiuvTHpB\nqKysFCZh8vLyKCkpEcjMeFi2bBmFhYW0tbXR19fH119/TVFREZ6enoSGhiIWi3nw4AGLFy/m5MmT\nZGZmTkmT87w1PPDHSa1HERERQUtLi0Ful5+fH1FRURw9elQQYT8K/Si1j48P7733HrNmzeLq1avC\nTcjjMDMzY+fOnZSWliKXyykrKwMekmexWMzFixdpaGjA09OT2NhYTp8+/UTyMTw8THx8PNHR0WMI\ngpWVFWFhYYJJ6fDwMMXFxXz55Zf84Q9/4MGDB8TExPA//sf/YMmSJcyYMYPh4WG++uqr52InYWpq\nyu7du/H29uatt95iYGCAxMREACoqKiguLjao/M6ZM4fi4mLq6+snTGvv7u4WyJxIJCI9PZ21a9e+\nbGVNAJ1OR0dHx8uW1kv8EWq1mqNHjxIYGPiTtSKXSCRYWFiQnJxMUFDQlC8AKpWKK1euMG/ePDIz\nM1m0aBF37txh+vTpz5xFo2+T6dObQ0JC6O3t5bvvvqOjowN7e3tB4zM0NERCQgJVVVVs27aNmTNn\njnm/K1euEBYWNiUhdEVFBWZmZvT29hIZGUl2djbNzc04OztTWVlJd3c3165dIzMzk2vXrlFfX49c\nLmfWrFksW7aM1atXs3DhQubOnYujoyNpaWksXbp0DLnRJ45PVZ8yVWRnZ6NUKqmqqiIzM5OIiAih\nRRcbG0tvby/p6encv3+f6upqduzYMeWE+OzsbGbMmEFlZSXXrl2jubmZlpYWOjs7cXd3H3NsyWQy\nxGIx6enp5OTk4OnpyZo1a0hKSmL16tXcvXuXjo4O1qxZg5+fH5mZmZSXl+Ph4TEpjVx3dzfV1dVj\nNETfB7du3WLGjBkG1ROxWIylpSXJyckEBwcLn1Nf7U1KSmL27NkGurT8/HxcXFyws7NDLBYzY8YM\nsrKymDlzJhcvXkStVuPg4GBQWTM2Nsbb25tr165RW1tLWFgYlZWVDAwMoFKpePDgAQsWLGDWrFm0\nt7dz+/ZtZs+ePYaw6nQ6vvvuO8zNzYmMjBz3czo4OJCWlsadO3dITU1FrVbj7+/PmjVrhFw0/fsa\nGRlx+fJloqOjMTIy4vr16890wyMWi5HL5Vy+fJnr169TVFREfn4+fX19giO3Vqvl7t27VFRU0NPT\ng7OzM5s3b2bevHnCeVhaWopCoTCYMEpKSsLKyoquri46OjowMTFhyZIlU97HFwXd3d2UlJQI02wv\nGl7S4McwOjpKfHw8jo6OUw7k/LFh1qxZ2NjYkJWVNeXXisViPDw8qK2txcTEhPb2doOYiWfB41Na\nRkZGLFiwQBhl//jjj0lNTaW6uppPP/0UpVLJnj17xr0b0Wq19Pf3j1vheZL5ob4yYGdnR35+PjKZ\njKKiIv793/+dU6dO0dLSIjjc/vKXv+Rv/uZvePvtt4mOjhYS2PUYGBgQ9E2P409R4RkYGCA7O5uK\nigpMTU3Zt28fs2bN4tq1a0RFRQEPRezd3d3U1NTwxhtvTNltVu+urG9v6BcatVpNXV3duFUijUZD\nW1sbra2thIaGsmTJEiorK7Gzs0OlUpGZmUlkZCQikQgzMzN27dqFlZUVn376KQ0NDU/dpx+iwvO4\ncFkPT09PbGxsyMnJMdgeFhZGQEAAR48eNajc6POf9CgtLcXKygq5XM4777xDd3c3+/fvF6aI9Jg2\nbRq7d++mu7ubS5cuCYGhejPNwsJCdDodK1asoK2tzaAyqUdhYSFNTU2sWrVqzHOtra2kpqbywQcf\nIJVKaW5u5q233mLr1q34+vqOOYZ7e3spKChgZGSEgwcPYmlpyb/8y79gYmIypcEDExMTZs+ezb59\n+/Dx8SE8PBx4eP729vYK01V6IXptbS0ffvghZ86cAR6SZ51Ox549ewgPDycrK4sPP/yQ/Px8Ghsb\nqaurY+nSpZSVlSGTyX4Q36CfE9rb2yctAfg54iXheQQ6nY7z588LWTQ/hx7wypUrycvLo6OjY8qv\n9fT05O7duzg4ONDY2MjMmTO/N+EZrzSuH2V/5513qKmp4YsvvsDZ2dnArO5xDAwMYGxsbEAqdDod\nJ06c4MMPP6SkpMTg9xYUFPDNN99QXFxMcnIyN2/epLGxEalUyiuvvEJ4eDgKhYLY2Fh2795NSEgI\n58+ff6JYcyIPHvhhCY9Op+P27dvs378fmUzG22+/zbJly5DL5Vy9epXZs2djaWnJvXv3+Pzzz1m0\naBGOjo6cO3duykShoaEBpVKJhYWFEDsAD9sTb7zxBmKxWJi0gYduzJ988gkjIyNs3LiRgoICNBoN\nxcXFzJ07l5KSEkxMTAwqdhKJhOjoaJYvX86xY8cEt+OJ8KckPADR0dHk5OQY5FIBvPLKK8yaNYuj\nR48yNDTE8PAwXV1dwoKib8kuXryY0tJSZDIZcXFxbN26lbKyMj766CPu3LkjfBZLS0vmz59PXl4e\nZWVlODs7U1pailgspru7mytXrmBkZMT69eu5dOmSoB2DPxKaDRs2COSlr6+P3NxcPvnkE2Haavv2\n7ezbt4/Zs2eTl5c37uetq6vj//7f/0taWpqwbxEREUilUnJycti5cycKhWLCdrI+dd7Kyop//Md/\npLCwkJCQEKqrqwkLCzOoCMbExBAeHo6xsTEeHh6Ul5fT3NxMcnKy8B0ODQ0JxGnPnj2sW7eOyspK\nDh06hI2NDRqNhpGREebPn49Go3mma92Lgra2thfal+hlS+sRZGVlUVlZybZt23704+eThbGxsSDK\nnDt37pRInEql4sKFC/j4+NDR0YG/vz8pKSlEREQ8U488OzubgICAMRMu8JCUnD9/nsHBQdatW0dd\nXR2XLl1CoVAwffp0g/0uKirirbfeIi0tDTc3N7y8vFCr1ezfv18wM2tqaqKwsJCUlBQyMjKoqqpC\np9Mhk8mYM2cOu3btIj8/nzfeeINZs2YJPjjffvstIyMjLFq0iFu3btHe3j7hKH5TUxPt7e3MnTt3\nzHNVVVVIJJJxW3HfB01NTZw6dYqqqiq0Wi2vvfaa0GLp6ekhMTGR119/nWvXrnH58mU2bNiAr68v\ns2bN4vr16zQ3N+Ph4THp4yA/Px8rKyvc3NxQKBRkZGQIJEulUqHRaDhy5AhVVVV0d3eTlpbG8uXL\niYyMxNramrq6OpqamigrK2P16tUkJCSwfPnycS+6NjY2eHt7k5qaSm1tLR4eHuOSxt7eXu7du0dQ\nUND3+zIfwf3795FKpePmwSkUCjQaDbdu3RozKeTq6sqDBw/IycnB0tKSlpYWQcxdW1tLeXk5q1ev\npq2tje7ublxcXFCpVMydOxcLCwtSU1MpLS3FxsYGMzMz7OzsuHbtmlA9HBoaQqlUsn37dpKTk5HJ\nZLi7u6NQKLh48SIBAQGMjo5y9OhRoqKicHFxoaysjNTUVCEPLyIigpUrV+Lu7i6cey4uLiQlJeHi\n4jJmcEAfKaMnVCKRiDlz5tDS0kJMTAxr167l17/+NTNmzMDMzIzBwUEkEgkqlUrQGs2aNYvLly+z\naNEiwTywrq6OuXPn0tzcLJgz3rx5k7q6OhQKBW1tbYSFheHj48Pf/M3f4ObmJkz2LVq0SNiXadOm\n4ezszPXr13F0dCQ5ORkTExOkUildXV1IpdIpT5O+KCguLsbKyuqpuYc/V7wkPP8/SkpKyMzM5I03\n3hh3Qf4pw8HBgdzcXIE8TBZGRkZUVlZiaWlJTU0NoaGh3L59GwcHh0lNVz2OjIwMwsLCxpTEGxoa\nOHr0KA4ODqxfvx5ra2t8fX2pqqri+vXrXL9+HTMzM2xsbKirqyMkJETw5vj2229xc3PjzJkzBjli\nQ0ND2NvbExwczKuvvsqyZcsIDQ2ltbUVOzs7nJ2daWpqQiQSCS0Ic3NzfH19yczM5O7du8TExJCS\nkoKlpeW4ZeDq6mqGh4fHHX+9d+8eMpnsuQne+/r6SE5OJj09ndDQULy9vamvr2fZsmUCeUlJScHe\n3p7bt2/T2NjIzp07hdFTiUSCt7c3V65cYXBwcNLjzRcuXGDGjBncvHlTmM7aunWr8L45OTlUVVUx\nOjpKS0sLu3fvNiCITk5OnDlzBldXV8FfZfHixRMSLqVSib+/P1VVVaSlpTFz5swx52Nvby8VFRUE\nBwdP+XucCI2NjWg0mgkXSkdHRy5fvsz06dMN2oIikQgPDw/q6+u5fv06Dg4OgoFjamoqPj4+ODk5\nMW3aNFJSUgTRtkgkwsrKiuDgYLRaLYmJiTQ1NTFz5kyhrVdXV8e0adOwsLAgICAAd3d3vv32W2xt\nbZk9ezY1NTXU1tZy584dRCIRQ0NDfPfddwwMDODn58eaNWuYM2fOuH5PRkZGqFQqLl26RGBg4Jgb\nmIqKCvr7+9FoNMjlcoaGhggKChLOFblcTmBgIHFxcfzqV7/ir/7qr/jLv/xL3n33XaKiorhz5w53\n794lKChI2LfMzEwaGhqoqqpCJpPR29uLubk5Q0NDmJmZ8eabbzJ//nyWLFkiHEM9PT2UlpaOidRI\nS0vD0dERR0dHOjo6hHR3mUxGe3s78+fP/1lU6J83cnNz8fLyemFFyy9bWjy8E0tKSpq0J8lPDfpw\n0QsXLkx5vNTT05POzk5aWlrQarVCrtZUoS9NP9rS0ul05OTkcOzYMVasWDFmTNrY2BiNRsPAwAAn\nT57k3/7t3zh48KBBm2lwcJDjx4+Pe3Hbtm0bYWFhBqV3fYYRgIeHh8EEDiBoSqZNm8axY8dYsmQJ\nZ8+epbOzc8z7T+SyDM+vpTUyMkJWVhZ/+MMfkMvl7Nu3j5CQEPLy8gxyp/Ri1nv37mFiYsKuXbvG\nHMvGxsZs27ZNMKZ8GnJycvjP//xPFi9ezI4dO9i2bRv/+3//b375y19SU1PD4OAg6enpgpBVp9MZ\nJNbDQ22KsbEx/f39ZGRkEBUV9dSFyMjIiLVr17JgwQKOHDlCcXGxwfPP23gQntzS0u/TypUrOX/+\nvBDy+ej+xMbGotVqaWhoYHR0lN7eXiorKwWvIHt7e8zNzSkvLzd4rVgsJjg4mF/96leCjqm/v184\nT5ubm4Xv19ramo0bN3L69GlaWloIDw+nsLCQ4uJient7sbGx4b333mP79u3MnTv3qR5cvr6+mJqa\nkpubK2zTt/W7urrYt28f7u7uqFQqmpubx/XBGQ/6ytCDBw94//33+e///m/OnDnD6Ogojo6O+Pv7\nMzo6ilQqxdHRkT179qBSqUhKShpzfRocHBzTBh8YGKC4uBhfX19SU1N57bXXiI2NxcbGBgcHB7q6\nuvjss88oLS39QQNQf4poa2t7oTU8L7zTcnt7OydOnCAuLu5nbcTk7OyMl5cXly9fHlfUOBE8PT3J\nz8/HxMSEtrY2ZsyYQWFh4ZR/v/6irSc0vb29JCQk0NPTQ2RkJN3d3Vy8eJG+vj56enro7e0VSIZ+\ngdFoNNTU1BgsODKZjNmzZ/P3f//3DA8Pc+vWLUpKSsboLcaDu7s7KSkpaLVagztciUTCqlWrKC4u\n5sKFC3h4eHDixAl2795t0Ors6emZsDSs1Wq/F+HR6XTcuXOHixcvYm1tbSDebmlpobW11SAK4+zZ\ns2i1WhYsWPDEVo9KpWL79u0cOnQIpVI5oU/TzZs3Wbp0KUNDQwJZ1SMpKYmgoCD++Z//WVi4LCws\nmD59upC1pceDBw+EUVilUjmlVkNAQAD29vbEx8dTW1vLypUrkUqliMXiP6mGRw8vLy9u3LgxJpEb\nHhIXY2NjjIyMOH36NDY2NmP8bubPn09ubu64xEEmkxEVFUVwcDAff/wxarUalUqFjY0NNTU13Lp1\nCz8/P6ytrfHy8uLAgQPI5XKhTbtt27YpG6OKRCJWrVrFwYMHmTNnDtOmTSM1NVWoDsrlciIjI/nm\nm28Et+aJMDIyQm1tLVVVVdy6dUvY3tfXh7u7O1u2bCE+Pp7c3FyGhoZwdXVl+fLlQsV527ZtXLhw\ngc8++4wtW7YILc/BwcExfkd5eXn4+PiQnp4uuKgDzJ49m76+PkQiEZ6enuTm5pKamioIzJ/FhPXn\nhOHhYQYGBn4SBro/FF5owtPf389XX33FkiVLfvDx4R8Dli1bxh/+8Af8/f0nnaljY2ODSCTC0tJS\niAHQL67j6Xi0Wi19fX309vYaPPRCwo8++oju7m7UajUymQwbGxvu37+PqakpKpWK6dOno1KpUKlU\n3Lt3j9TUVCQSCZ6enqxevRqxWEx5eTmFhYWIRCIsLCz4q7/6K+DhohEcHPzEVsejFR4zMzPMzMxo\naGgY13127ty5TJ8+nRMnTiASiUhMTCQuLk54/oeq8LS0tHDhwgX6+vpYtWrVGM+b3Nxc5s2bh1Qq\nFdohdXV17Ny5c1KaIUtLS7Zu3cqXX36JQqEY8xqdTkdcXNyEBGB0dJSuri4+//xzLl++/EQDs6Ki\nIvz8/Lh9+zZqtXrKRNDW1pa9e/dy9uxZPv/8czZs2PAnFy3rIRKJWLlyJQcPHsTPz8+ggqbVamlr\na+M3v/kNJ0+eJDs7m927dxu8Xp/99KRMK5lMhkajwdLSkr6+PmpqaggKCuLcuXNkZ2cLQmM7Ozu0\nWi3R0dGMjIxw+vRpQUQ+FVhaWhIeHs758+dxcHDg3r177Nq1S2g7K5VK+vv7x9hz6HQ6mpubqaqq\noqqqivr6eqZPn46bmxvW1tb09PQIPzc0NERKSgplZWUYGRnx5ptvjvn8YrGYmJgYIffu9ddfx9XV\nVQgO1UOtVnP9+nUiIiJobGxk06ZNwnOzZ8/m2LFjmJiY0NnZye7du6mrqyMnJ4crV64QHBxMaGjo\nM3l3/RzQ0dGBhYXFC+1R9MISHo1GwzfffMOcOXOeq/jxxwz9NNRUMnX0d0utra0UFRVRVVUlGHxJ\nJJIxxGZgYAClUimQFv3DxsaG2tpa3NzcKC4uZtOmTU8Mq4SHhFSlUrFmzRqDRT8nJ4fMzEyys7Pp\n7+8nOzubJUuWPJM5mr6tNZHdvp2dHXv37uX06dOUlpZia2tLREQE8JDwTNQCfRbCMzAwQFpaGqWl\npURGRhIcHDzmPfRGg7/61a8YHBzk9OnTNDQ0EBUVNSWBtL29PevXryc+Pp4dO3YYjFKnp6cLyfIT\nQavVcvv2berr6yesEmm1Wm7dukVoaChmZmYoFIpxqyNPg1wuZ/369Vy7do2DBw+yaNGiPwvhgYcE\nISQkhIsXL/L6668L29va2jAzM0OpVBIQEEBjYyM3btxg1apVAsEWi8WEhoaSl5fHa6+9Nu7719XV\nYWtrS0tLCyKRiNHRUa5fv45UKuWzzz6jpKQErVaLSCTCzc0NFxcXduzYQXV1NVeuXGHx4sVT/uwR\nERH813/9F01NTbzzzjsG59GdO3cQi8VIpVK6u7uprKykurqaqqoqFAoFbm5uzJs3jw0bNgjVrOTk\nZMRiMY2NjUJ2XWNjI5GRkeTl5T3RtT4kJESI8IiKimJ0dNRgfwoKCnByciI7O5vt27cbnB82NjZI\npVKsra0Fw1VnZ2ecnZ1pb28nNzeXDz/8UBiPn2pu3k8dL3KGlh4vJOHR6XQkJCRgbm7+wplU+fn5\ncfPmTfLz85k/fz46nQ61Wj2GuOgfnZ2dNDc3G7yHTCajsrISd3d3bG1thYgGlUqFqanpuETq7t27\naDQa6urq2Lt376TMC2fOnMlvfvObMdvFYjGRkZFERkaSmJjI/fv32b9/PxEREYSGhj51wu5RDYmH\nhwepqalPXCgUCgVbt24lOTlZmHwJCgp6bmPpo6Oj5Ofnk5mZia+vL/v27ZuQvOXn5zNnzhx6eno4\nfvw4jo6OGBkZCSRsKnB1dWX16tUcO3aMXbt2CW2E5OTkSfmZ6HQ6QZg7Hu7fv49SqaSwsJCVK1di\nYWExbnVkMhCJRMyfPx9HR0eOHz/O6Ojocx39nyzhgYfj6B9++CHV1dVCi+5R/52bN2+ycuVK8vPz\nSU1NNRCWBwYG8v77709YHXw0nVwsFiORSNi3bx9nz56lqKhIMOoDqKmpobi4mMTERNatW8eBAwdw\ndXWd8mTgzZs3gYd/z8d9prKysjAyMuLjjz9mZGQENzc33NzcWLZs2YStkejoaMrLy6mvr6e5uZnp\n06ezfft2VCoV5eXlNDY2PnFKyNXVld27d/P1118LU2nwsG2WnZ2NlZUVISEhY4iTSCTCx8eHzs5O\nent7UavVQqXKysqK1atXExUVxfXr1zly5AiOjo5EREQY+Ev9nNHW1vaS8Py5d+DPgdTUVPr6+tix\nY8fP+kDXaDQG5EWvjTEyMiIlJYXc3Fz6+/sFA7hHSYuFhQUuLi6YmpqSkpJCY2Mj8JDshIWF0dLS\nMmmyWFtby5kzZzA2NubNN998rv40S5YsYf/+/bz++utcv35dMN3z9/cfl3g9XhlwcXGhvb2d/v7+\nJ07niUQiYmJiMDIyIjExkZ6eHoaHh8fNU4LJE56KigouXLiAubk5u3bteuJdp0aj4caNG0RERHD0\n6FGio6O5ceMGUVFRkw48fRyzZ89mYGCAL7/8kt27RU4qWwAAIABJREFUd2NqajppYbtWqzVYgB9H\ncXEx06dPF6ZoRCLRuNWRqcDJyYmNGzdy9OhRvvjiC9avX/9cMoGmQniMjIyIjo4mKSmJd955B4lE\nQlNTE3Z2djx48IC2tjb8/PyYNWsWhw8fRiaTCc7HCoUCX19f8vPzhfNncHCQ27dvU1xcTENDA8bG\nxkRGRqJQKDh37hympqZIJBJkMpnB9z04OEhlZaXQFjY3N+fYsWP4+voKBoFyuRxjY2Ph34///9at\nW2RmZvLmm2+SlZVFQkICtra2VFVV0dzcjEQiYdq0afj5+bFgwYInXi91Oh337t0jLS3NYGJy8+bN\nwt/I1dWVqqqqp45FW1pasmfPHj7++GNKS0sJDw+nrKxMiADRj6k/jtmzZxMfHw88rE49bhlhYmJC\nZGQkERERFBUVcfbsWWF038fHx+Ca8Wj7++eAJ1lsvCh4IQhPX18farUaKysr8vPzuXPnDrt3737m\nReLPjdHRUQNxr/6h187ot4+Ojo5pLalUKhwcHFAoFAwMDPDuu+8+1Tl1+/bt/Od//qdwVx0YGMin\nn3761AuCTqcjKyuL3Nxc5s6dS39//3M341MqlYSGhgptsrq6OlJTU8nJyWHp0qV4eXmN2cdH/6/3\nyqmqqhIytp6EZcuW0d/fT05ODhKJBI1GM64Y8nHCo59+8fX1ZcaMGTx48ICUlBQ6OjqIjo7G09Pz\nqRfXwsJCweJ/586dQgvx+6aGh4SECHq2N954Ax8fnzGL63iQSCRYWVmNexwMDw9TVlaGSqVi5cqV\nwvP66sj9+/ef2aNIqVRiamqKu7s7n376KXFxcd/7Qm5sbCxojCbT6vX29ubGjRtcu3aN8PBwmpub\nWbBgAfn5+QQFBSGRSFAoFOzYsUMgPXqX4fnz53Po0CFsbW0pKSmhuroaFxcXwsLC+PbbbxkdHWX+\n/PncuHEDU1NTSkpK8PLyGjNxJJVKWbx4MevWrePbb7/F3t4eY2Nj6uvr8fX1ZXBwkK6uLtRqtfAY\nGhoy+DcgVHA0Gg3w0OvJxsYGpVKJvb09w8PDQrTDeKRJH2yalpbG0NAQtra2VFdX8/rrr/Ptt9+S\nm5tLTEwMAG5ubmRlZU1IWB7/mzg5OaFWqzl48CAjIyMMDw+zY8eOCa8jdnZ26HQ6zMzMBLPL8WBk\nZERISAjBwcHcuXOHnJwcQeAcGBiITCYjMTERrVbL2rVrfxbEp729fdzA3xcJP80Vf4rIyMigoKCA\niIgIbt68yZtvvjnhnfmfE/q4hInaS/rH0NCQIPJ9tCozc+ZMg21608Hx4O3tzUcffURtbS2enp5P\n3C+FQkF4eDhXr14VkpwVCgWtra0TTrb19/eTkJCAWq1m79693Llz53sFjz4J4eHhvP/++7S2tuLs\n7MyuXbuoqKjg0qVLZGVlsWzZMsEPZzzth17HMxnCAxAbG0t9fT3d3d0cPHiQjRs3jhn1fJzw3L59\nm4KCAu7evYu3t7eQZ7N58+ZJkcDe3l5SUlKE9oCxsTEJCQksXrz4e4sQR0ZGCA4Opr29XXC5noxG\nRiqV0tbWxscff0xQUBB+fn7CeVVWVoaVlRVisdhAf2VkZMSKFSuE6siz7Lv+mF60aBFOTk58++23\nzJs3b0qBqOO9p7GxsWD0N5mfX7lyJZ9//jlz5syhubkZS0tLbt26xS9+8Qvh50xNTQXSI5VKsbe3\np6ioiKGhIRITE9FoNILNgF53YmVlxeDgIO3t7bi7u5Ofn8/evXvZu3cvBw4cED6jp6cng4ODFBYW\nsmnTJr755hvWrFnDlStXkMvl45KK3t5eqqqqKCoqorq6WrhuODg4CBNhxcXFODo6UldXx/Tp02lo\naKC1tZXS0lIDwqRWqxkcHBTOa33CfHd3Nw4ODhQWFiKXyyksLESpVArO6A0NDdy9exdTU1ODypNU\nKh3z9xsaGiIsLIzy8nJu3LjBnDlznpgFp29r1dXVCSakT/s7ent7C75W2dnZXLlyBT8/P4qKihCJ\nRNjb2xMaGvrU9/oxQx+G/CKPpMMLQngqKioYHR0lMzOTsLCwP7m1tk6nY3Bw8KlEpr+/H4VCMaYi\n4+joaPB/fb7O94GRkRGrVq3i3LlzvPfee0/VvYSEhHD58mUGBgY4duwYra2tlJWVjUt4ampqOH36\nNL6+vixZsgSJRDKun8bzglwuJyIigvT0dDZu3IhIJMLLywsPDw+Ki4s5ffo0dnZ2LF26FGDMd+fu\n7k56evqkS9gSiYR58+aRkpKCu7s7hw4dIjY21kDL8ijh6e/v57vvvkOr1dLT00NTUxPvvffepA0u\n6+vr+frrr5HL5ezevRuJRMLt27cRi8UT6mcmi4aGBg4ePCgsNhqNhqamJrZu3Up8fPy4Cd/wsMry\nt3/7t/T399Pf38+FCxdITk7G2NiYtWvXUlhYSH9/P2vWrBnznfr4+HD9+nVBRzZVPDql5ebmxt69\nezl16hR1dXXExcU9882Mvq012ddbW1sTFBREUlISUqmUyspKXF1dx+iTdDod3t7eJCcno1AoUCgU\nwvSSWCzG3NwcsVgsEIfGxkaKiooEA7179+6xf/9+LC0tOXv2LBcvXmTFihWEh4fzzTff0NDQwNdf\nf01QUBBnzpxh3bp1nD17lhkzZgimoXqxcU9PD9OnT6epqYkNGzbg4+Nj8Pdxd3enubmZyspKwsLC\nWLx4MTU1NVy6dIkNGzYIP9fU1ERaWhotLS0sWrQIExMTzp07h7e3N/7+/mg0GoaGhrCwsODq1asM\nDw/T39+PWq0WQknFYrFB1Umn042pHjU3N5Obm0tdXR1GRkbcvXuX+Ph4vLy8MDY2HtOuk8vl+Pj4\nUF5ezvDwMJ2dnZPOj9O3Szs6Ovjkk08E+4uUlBRsbW0nbdb5Y0R/fz9SqfQHuwb/VPCzJzwDAwPC\niCQ8HOm1sbF5bpNZTxL8PvowMjIao5OxsbHBzc1N2G5iYvKDB04+Cg8PDxwdHcnIyBDIwHgoLy/n\nH/7hHzh16hRyuVxYcH7/+9/zxhtv8Hd/93dCVUCfMr5u3TqDytHQ0NBz0VpMhNDQUN5//30aGxuF\nO0CxWExAQICgmTh8+DByudxgIgnAwsICY2NjmpqaJp0krtFo8PLyoqSkhHXr1nH+/HkaGhowMjLi\nd7/7HaWlpWzfvp1/+Id/4NixYwbtoebm5klXNm7cuCGMfoeHhyORSNBqtaSlpRETE/O9ia+9vT3T\npk0TfIv0LsDR0dE4OzvzH//xH4KwHRAW69///vf88pe/5NChQ9TW1goERCqVkpaWxoMHD1CpVOOK\nJPV6qMOHDwt6k6ngceNBMzMzdu7cyaVLl/j0009Zv379M1nnT0XHo8eiRYv4r//6LywtLcnPzxda\nN0NDQ5SWllJcXMyDBw+YM2cOcXFxJCUlodFokEgkjIyMIBaLUSqVWFlZCYZ9M2bMICIigry8PPLy\n8oSqr4mJCa2trULFVd8yS0hIoL29nYqKCkxMTEhISMDJyYnPPvsMkUiEg4MDbm5urFmzhtHRUU6c\nOMHWrVvHbSmKRCKWLl3KJ598QnR0NPBQT6O3lXjw4AHp6enU1taycOFCNmzYQF5eHmlpaWPOeT3y\n8vIwMTERhPUZGRkMDg4K76/HyMjImPbbqVOnhGiPoKAgdDodpaWlNDU1YWVlxfDw8JhWnUQiYXR0\nFJ1Ox6FDh7C2th6XGE2kb3q8nTs6Osrhw4eJjo7m97//PQsWLOAv/uIvpnSc/LnxUrD8ED97wlNR\nUSF4f0gkEoKCgsaYo40HjUYzRhPzqE5G/9D3ix8lMubm5jg7Oxts+7Fmc0VHR/Pxxx8zd+7ccQWz\n6enpxMbGMjg4iFarFfr8ehw6dIj4+HiSkpKoqKhgZGSEt99+e8xd7tDQ0A86BmpkZMQrr7xCWloa\n27ZtM3hOKpUSHh5OYGAgn3/+OWlpafT19bFw4ULhjkff1pos4ent7cXJyQl7e3syMzPZs2cP//Zv\n/8Z//Md/CBfL//f//h937twhICAAY2NjlEolEokEtVpNTU3NE8fyR0ZGSEpKora2ljVr1nDu3DnB\naLCwsBCVSvW9dStarZaCggLUarVg5mdraysc74sXL+bdd9/lk08+ISsrC7FYzLJly9izZ49QGl+/\nfj0ffPCBcFw4ODgwffp0uru7sbOz48CBA9jb2xMYGIi3t7egm7OxsWHu3Lmkpqaybt26Ke33eMaD\nEomEFStW4OLiwtdff82iRYsIDQ2dEiF8FsIjk8mYOXMm1dXVmJiYMDw8zMmTJ7l37x5ubm6EhYXh\n6ekp3MhYW1tz9OhR7OzsaGpqYnR0VHiPBw8eIJPJ2LJlCyMjI/T19RlMy1lYWFBZWYlIJKKgoIAF\nCxYgkUiIiori/Pnz1NfXo9VqGR0d5f79+zg4ODBt2jRhBL6pqYlvvvmGuLi4J+qnqqursbOz48qV\nK2zbtg1TU1PUajUnT56kurqaiIgIXn31VTQaDfHx8QwNDbF3794Jp7acnJwoLi4WCI+bmxuJiYlj\nfk4qlSKVSg0IsEajoaqqCnd3d2JjYwFYsWIFp0+fRq1Ws2nTJoOKnE6nQ6PRcP78eSHeZcGCBWOI\nVG9vL21tbRPqmx6HSCTil7/8JQUFBVy4cIHIyMgx3kQ/ZrwcSX+Iny3h6e7u5vTp04Jx1YoVK/D1\n9UUkEtHX10d9ff0Ywe+jxEaj0Ywr+LWzszP4/9MEvz92qFQqIiMjOXfuHG+88YbBAnH37l1iY2Pp\n7++f8PUjIyN0dXWxZMkSvvjiC1577bVxqxdDQ0MGrrM/BIKDg8nOzqa2tnbcDCtjY2Ps7OwIDg6m\ntbXVYJTdw8ODjIyMSYkp4Y+EZ86cOdTX13P16lXy8vIM7gz1gaj//d//TWdnJ9nZ2bS1taHVaids\nFcFDB+cTJ05gZmbGW2+9RXJyMqGhoUJV4MqVK4IB37OiqqqKCxcuoFAo2LlzJxcvXhT0XD09PSxb\ntoxDhw5hYmLCP/7jP074PiqViuXLl3P+/Hn8/f1RKBRcvXoVExMTNm3ahFarpby8nIKCApKSkvD1\n9SUoKAhbW1uioqLYv3//E434xsOTjAe9vb2xtbXlxIkT1NbWsnbt2kmfo89CeHQ6Hf39/QwPD6PR\naMjKymLu3LmsXr163PaBnZ0dW7Zs4dixY/j4+FBSUsLo6KjQZn/ttdeQy+VCBePRmInW1lahzXL1\n6lXa2tqorq5mdHQUd3d3vL29qa6uZu3atZw/f57GxkY6OzspKSlh+vTpfPXVV6xevfqJJqtarZb8\n/HxeffVVzp8/z7Vr14Tfq1Ao+PWvf41cLqe2tpZTp04ZtK0nQkhICKdOnRIE4Q4ODnR3d9PX1/dE\nE8DR0VGGh4cZGRlh/fr1wna5XM6mTZu4fPkyBw8eZPPmzYJjs0gkQiaTERAQQHV1NZ2dnbi6uk5J\nK6bT6finf/onYUzf3NycefPmcevWLSEa5KemhXlZ4XmIny3h2bZtG6mpqUilUj755BPy8vJITU1l\ncHAQExOTMURGn2KsfygUip+FMn8yCAkJobCwkKKiIgICAoTt//zP/zzpEeXR0VGKi4sNLkyP4vEc\nrR8CEomEyMhILl++PIa86aHT6VAoFKxZs4bw8HAuX77M/v37WbhwIc3NzZPWGuk9eEQiEa+++ioH\nDhygpqZm3H06ePCgwb6IRCIKCwtpampCqVSiUChQKpUolUq6urpIT08nJCSERYsWMTAwIBgNwkMf\nHnt7+2dOO25vb+fixYu0tLSwfPlyQcMRFxdHV1cXaWlphIaGYmJiIkRQmJiYGMRYPA79lFdYWBjt\n7e1cu3YNpVLJZ599xqpVq/Dz88PPz4/Ozk4KCwsFN9zAwEAWLVrE+fPneeuttxCJREJg5ng+PTqd\njmvXrnH69GkyMzPp7e0lLi5uzJSahYUFe/bsISkpiQMHDrBhw4ZJxcboJxcng+7ubm7duiVobUQi\nEVKplI0bNz7VyVevFYmPj8fY2Jjdu3dTX1+Ps7Mze/bsARCI8dDQkEDwHifTEomEbdu2YW1tLRxf\nJSUlJCQkEBcXR3p6Os3NzSQkJCCXy4mOjn6q5quiogKFQoGFhQVWVlYkJycTFhaGh4cHrq6uyGQy\nsrKyyMnJYe3atZOqlusrmeXl5cyePRuxWCxUxZ40KKA3v5w3b94Y0qqvNtrY2HDkyBHWrVtnsC8u\nLi5oNBp0Ot1TfX8eh06nQ6lU4ubmxoIFC4Rj54svvuD06dP4+fk984Thnwvt7e0/aQ3S88LPlvDo\nGb1EIiE4OFiIEVAqlS+0tfZ4EIvFxMbGcuzYMby8vFAqlXR3dxMfHz8mKHEiDA8Ps3//fv7u7/5u\n3Pbd4ODgD17hAfD39ycrK0sog48H/eKgD2Osr68nNTUVeHjn/KhR3ER41GVZoVCwceNGoa3wKEmU\ny+X89V//NZmZmeTl5QliZjc3N5RKJQMDA3R2dgombZ2dnRgbGwtu0vpW7IkTJ5DL5cL4fHZ2tkCS\nHiVME03mDQ0NkZGRQWFhIREREaxfv97AlkGlUmFkZERDQ4PQKjM3N2fr1q0cPXoUhULxxO8zKioK\neNgCNTEx4d1336WkpITjx4/j5eXF0qVLsbCwYPHixURGRlJVVcXNmzeprKxEIpFw8eJFli5dyjff\nfINUKuWdd94xaFVkZGTw1ltv0dDQwODgoEB+/v3f/10Qjj8aKSKVSlmzZg1FRUV88cUXLF++3IDM\nj4enVXjUajVlZWUUFxfT3NzM7NmzWb58OcePHxdGmS9dujSpFt3MmTORSqX8r//1v4R2YEtLCyEh\nIZSWlpKeni5kyelJularxdjYGFNTU8Eb5/E2sT4QND4+nuXLlwtVRf13Np4wX6fT0dXVhYWFBTk5\nOZiYmAgRNP7+/gwPD2NtbS24rQ8MDDyxhfU4xGIx06dP5/r168yePRv4ox/PkwjP+fPnAZ7o9+Xv\n74+lpSXx8fGEhYURHh6OSCQSBP23bt2isLBwSoRHLBbzl3/5l2O2KxSKMe3ynwpetrQeQvLb3/72\nt3/unfghEBsbi5eXF//6r/+Kp6cnqv+PvfcMjvI89/8/21RWva56QV1CHQnRZHrvGGMbg4OTOHHK\nyZwzefefyZzMOWd+4+MZn4zjJCdxPI5tcMGmCHUQQh0VhLpQ7733lVar3f8LZu8gJEDgcuJY3xle\nANpnn3320f1c93V9i4UFRkZG35uuzdPCwsKC8fFxmpqaCAgIIDc3V8zJVwq5XM6xY8dEe/lB5OXl\nsW7dum+8yyORSFAqleTm5hIVFbXk+66trcXe3n7Rjt/S0pLw8HDGx8eprKykoaEBOzu7Ry7oBofh\nHTt2iOLZ3NwcPz8/CgoKGB8fRyqVYmtry89+9jPGxsbYu3cvERERDAwMMDw8zMGDB/Hz88PLywsv\nLy8aGxuZnZ3l3Llz7Nixgy1bthAbG8udO3c4evQoPj4+9Pf3I5PJcHNzY2pqisHBQdrb28XIKC8v\nj1u3blFcXExZWRnV1dXU1dVRXFxMRkYGGo1GWPfPzs6KjoFBoVVXV8fc3NyiwsDc3Bw3Nze+/PJL\nvLy8HuuQfPPmTf70pz/x6quv4ubmhkqlIioqio6ODlJSUjA2NsbZ2VlcG0Osi0ajobCwkNLSUubm\n5tBoNDQ0NBAeHo5MJiMlJYXDhw/T39+/iENm4GsMDAxw/vx5tm7dumSU6eTkhJ+fH6mpqfT19bFm\nzZpHjl/6+/uZmJhYRLzV6XQ0NzeTlZVFcnIyWq2WyMhIDh48SGBgoBgZHTlyhICAANLT03Fzc1tR\nMXDq1ClR1Bg+jyGLzgADX8nOzo6tW7fS2trK+vXr2bVr1yPdyq2trfHz8yMhIQG1Wo1MJsPCwoKG\nhgZaW1txd3df9Hs4PDzMu+++S1FREcPDw3h4eHDy5ElCQkLw8vISo8/y8nJ8fHw4fvz4U/8eazQa\nKisr2bJlC3B/vJyVlUVcXNyya7LhvjUzM2Pz5s2PPbaVlRXBwcFkZmbS3d2Nr6+vcKmuq6tjYmKC\nuLi4pzrffyYsLCyQkZHB7t27v/eb/X/aDo+FhQVnzpz5vz6N7xS2b9/OH/7wBzo6OpiamnrqvCKZ\nTPZIvs+3MdIyICQkhLy8POrr65clBi+3wEokEjZu3EhDQwPR0dFcvnwZlUrFjh07lhRw09PTwjfk\nQaxfv57/+I//oLe3l4aGBn7zm99gZmbGtWvX+Nvf/saLL77ImTNnGBwcFLut0dFRPv/8c1Qq1ZI0\n9traWtzd3QkICGBmZoaWlhZ+9KMfPdZWYWFhAbVaLVx4CwsLkclkREREIJfLRZE0MzODWq1mZmaG\n2dlZjI2N0el0KJVKPvnkk0WdI1NTUyIjIzl//jxHjx7FxcVFELDh/ndbX1/PkSNHmJ6e5l//9V/F\n+ZiYmLBv3z4iIyNJTk6mrKyM/fv3C3K4Uqlk9+7daDQa6urqREdxaGhImAq+8MILT+TWzMzMcODA\nAbq6upaMlBwdHfnxj39MUlIS77//Pi+88MKy19DU1FR05/r6+qioqKC6uhorKyvCwsLYu3fvEsl6\nVVUVZmZm4h4x8JkezKrr6uoiMTERFxcXQXR3cHBY1ifG4Kuzfft2EXTZ2dmJVqtl7dq1JCUlrajr\nahBK6HQ6fH19aWxsFITc9957j40bN7JhwwYWFhZEtIRBJl9XV0dERASWlpYYGxvj7e1NZWUlDg4O\nS5RVK4XBYdugojRc/+Hh4SV8GLVaTVJSEnq9ftnN03KwsrLi3LlzXL16lY8++ohTp07h7e3NwsKC\ncEX/vqalj46OYmlp+a0qgP9R8U9b8Kzi6WGY8ycnJ69oNv8w5ufnl911GiIIvi2Ct0QiYdu2bWRm\nZuLv77/ELv5RsLW1RS6X4+TkxC9+8QtKSkr48MMP8ff3Z+vWrWLX/rjQ0P379/P+++9ja2uLkZGR\n6Hrl5eUJgqUhA6ipqYmrV6+Kbs6DhZher6ewsJD9+/cD9ztkISEhT/SQkslkaDQaMjMz6evrY9eu\nXQQHBz+2s2kgUb/77rvs2bMHiUSyqCAaHR0V3LeLFy9iYmKCWq0WEnYDofXIkSO0tLTQ3d1NSUnJ\nolGbUqnkzJkz1NTUCMLug4Gvzz33HKWlpcjlctHpGB0d5c0331zxWFWr1XLhwgV+8pOfLPk/Y2Nj\njh8/TklJCe+///4S3yQDenp6+NOf/sTc3BxhYWG8+uqrjyWoNjc3LzrO2rVrKS0tpbS0VLjaGhsb\nMzQ0xMDAAJWVlYK86+PjQ1NT06LjGRkZYWFhgV6vp7KyEo1Gg6WlJY6OjmRnZ2NkZERDQwNRUVGP\nVH7Ozs5y/vx5goKC2LRpE59//jkmJibMzMzQ1dXFuXPnyM7OpqioCK1Wi5eXF1KpVEj9jYyMsLKy\nQq1Wk5CQwOTkJK6urgwNDT35S3gEDMe8ffs2J06cEOGnLS0tS65vWloaFhYWqFSqp7KyMDIy4uTJ\nk2RlZfHee+/x0ksvERAQQHV1NY2NjYSEhDzz+X+XsWo4+HesFjyrWITg4GDKy8uXlf4+Cba2tmg0\nGuHIqlarmZubIyoqSvj3fFvw9/cnNzeXmpqaJTyBR52HRCIR8nSVSiWk7Pn5+fz5z38mIiKCLVu2\nPDY01EBcNXgC+fj4IJFI2LJlC3Z2dpw/f54DBw4Icu/JkyeXJRM2NTUhl8vx8vJiYmKC8vLyRQ6+\ny2Fubo6cnBzKysrYsGEDJ06cWFF8ilQqZWhoCBsbm8dK5efm5khMTKSxsRGFQoGHhwdr1qzByckJ\nrVbL7t27qampobGxkdHRUfHvDxZPEokEU1NT4Tzt7OyMi4sLJiYmeHt7MzIyws6dO7GxscHMzIzA\nwMAVE+enp6f53e9+JwoetVpNaWkplZWVvPrqq5iZmREbG4urqytffPEFHR0dxMfH09jYSGVlJZ2d\nncjlcvGdPOl+7e/vR61WLxoBGjyGPvroI9asWUNfXx9NTU2imDAkncvlck6fPs1bb70l+DVyuRwr\nKyteeeUV2traCAsLo76+nvn5edGpWFhYoKWlhTfffBMrKyscHBwW/bG0tOTzzz/Hzc2NHTt2IJFI\niImJ4dKlS+Icv/jiCyQSiRhjm5ubi/Nydnbm9OnTjIyM8OGHHxIQEMDJkycZHx/n97//PQMDAyvu\nujyMwMBAqqqqxN/XrFlDbW0tMTExYkPU0NBAR0cHarUaX1/fFRe7D17/bdu24eDgwEcffSRiPqqq\nqr63Bc8qf+fvWC14VrEIhgX7r3/9K2fPnuW99957YqYS3B9NnDt3TiysBoM8uL/QfRuE5QchkUjY\nvn07SUlJBAcHi3buk4o4X19fioqK2LRpE3B/JLNjxw5iY2PJzs7m3Xffxd3d/bFmeTY2NkilUhIS\nEvjJT34ifjY4OBilUsmFCxdQKpX86Ec/eiTXo7CwUPAbsrOziYyMfGSRpdPpKC8v59atW/j6+vLG\nG288tcljQ0PDsl29ubk5GhoaqKmpoa2tDU9PTzw8PFCr1bzwwgtLugzW1tY0NzczNTVFW1sbmzdv\nZvPmzchkMsG5mZmZEd2G27dv09jYiI+PD3Z2dvT29pKTk4NMJmNmZob+/v6n+hxdXV20tbVRVFQk\nuicGbxoDnJ2dRfhnUVERnp6eREdHCwLyShU4RUVFAMLIUqfT0dPTQ1NTE1KplD/+8Y/4+Pjg4+Mj\nYisAobh67bXX2LdvH2+++Sbt7e14eXkJXpNarWbv3r3s2bOHwcFB6uvraWhoEIG1/v7+REdHMzEx\nweDgIA0NDUKqrlAoMDU15ebNm9ja2pKcnLzovKempoiKiuLQoUOo1Wpu3LgBgEql4tVXX6W0tJTc\n3NxFXTBbW1uUSiWJiYm89tprz7R5MZgpTkxMIJVKmZmZoaGhgTfffBOpVMovf/lLkpKSWLNmjdhw\nPesYfO3atdjY2PD555+j0+lEbMf3EUNDQ0+DyE6RAAAgAElEQVRl+/DPjH9a0vIqnh2mpqYsLCwg\nkUgoLCx8olRXIpHg7u7Opk2bBKnUUFgY5J1dXV3fulGXjY0N9fX16HQ6wRmpqalBpVI90gTR0tKS\n9PR0YmNjF3VHjI2N8ff3JyAggJKSErq7uzEzM0OlUi2resnKyiI2NlYEp0okEoaGhrhy5Qre3t6C\nbOvn57eESNjf309hYSGHDx9mdHSUGzdu8Pzzzy87wmhra+PixYsMDg5y7NixZSW8K0FqaipbtmzB\nwsJCqJGysrJITU1lbm6O4OBgDh48SGRkJKGhoXR2dlJRUSFkxgYYzAx1Oh06nY62tjbc3d2xtbVF\nIpEgk8kwMTERkSmxsbEYGRlRUlKCk5MTW7Zsoaamhtdff51NmzYJP5SVwsTEBBsbG/r7+8U56PV6\nGhoauHv3LgUFBWRmZtLU1IS9vT1WVlb09fVhaWmJVCqlurqawMBAMXYyjO0exuzsLNeuXcPKygpj\nY2Py8/NJTk6mra0NKysrYmJiaGtrE0WDkZGRKMAkEglhYWHExcXh5ubGqVOnOH78OAMDA9jY2KDR\naDAzMyMgIACJRIKZmRmenp5ERUUJn5+amhrKy8tFVERUVBQ9PT2oVCqOHTuGlZUVs7OzdHd3C5+g\nB1FRUcFvf/tb/uVf/oUrV64It+c7d+4wMjLC2bNncXd3X/SatrY2xsbGUCqVK5L5PwxjY2OKi4vR\narXk5+dTUVEhilF7e3v6+/uxt7enoaGBw4cP09LSgq2t7YrNQB+GpaUlISEhlJaWotFoCAsL+15G\nKxQUFBASEvJIkvv3CasFzyqWhaurKyUlJbz22mtkZGSwsLCwbPinYaH84IMP2LVrF3fv3l30gDLI\nUefn53F2dsba2vpbHW3Z2dmRmppKTEyMeKA9ruCRyWQiVHG5ubdSqaS3t1coqwoKCrCwsFjkhaLT\n6cjLy+Ps2bNUVVUxPDyMRqPh888/Jz4+nu3btxMeHk5tbS0lJSX4+/svIlTevHkTX19fvLy8SE1N\nFblgD2J0dJTExERKSkrYtm0bu3fvfqyC6nEYHh7mzp07uLi4LCpygoKCRJHj5OQkCkCJRIKfnx81\nNTW0traKB7Ph+hk6NHDfD+txRneG6IPIyEhxPQ1RBr6+vly4cEHEGqwEERERvPfee0ilUrq7u8W9\naGZmxszMDO7u7oSFheHr6ytsKhQKBdXV1fT09DA9PU1LSwvFxcVkZ2eTnZ1NcXExd+/epbKykpqa\nGsrKysjOzhZj24mJCaytrYmOjiY8PBx3d3dsbGywtLQkJycHJycnsrKyMDU1FdEHL7/8svjODdyj\n+Ph4oqKiyMjIwN/ff8moc35+nry8PM6dO0d5eTknTpzA1dWVjo4OUlNTmZiYwNvbG2tra7y9vcV4\n2eA7tWXLFtzc3Ojv7+e///u/RSSIVqulqamJ8vJy4uPjmZ6eRqlU4ubmtuh3tbu7G3t7ewoLC4mM\njHwm9/ienh5aW1s5efIk5eXlovPm6uoqLBHkcjkxMTFCTv5VHNqNjY2xsLCgvr6e+vp6QkJCvvNm\nsU+LjIwM4uPjv3efezmsFjyrWBYG74zc3Fz++Mc/YmtrS3V1NTqdDlNTU6RSKe7u7vz2t7/ljTfe\nYHJykujoaNRqNX19feJBc/DgQdasWcPg4CBNTU0UFxej1+uxt7dfEb/kq8LS0pK2tjbUajVubm7U\n1NTg5OT02EV0amqKnp6eRxK379y5Q0hICNu3b8fGxoZbt25RWVmJnZ0d1tbWaLVaCgsLiY+PZ82a\nNVy5coWmpiZefvllAgICgPuFQXBwMGNjY6SlpbFmzRrMzMyYnJwkNTWVY8eOMTw8TG5uLidOnBAF\nxNzcnJBIBwYGcvz4cZycnJ6piDQoo1JSUoSSJSgoiAMHDhAVFbWoyHkYUqmUwMBACgsLGRoaEh49\nMpmM8vJyIiMjUSqVLCws4O3t/cRzUSgUTE1NIZFI6OrqorW1lfb2dtzc3ETB/CRYWFjw1ltvERAQ\ngFqtZnp6momJCWQyGceOHWPfvn2sXbsWd3d3XFxcBAcpNDSUyMhIWlpamJyc5Ny5c+zcuZP4+Hg2\nbtwoSL0TExN0d3cD9/lBFhYWODs7o1KpmJubo6enh/r6eioqKiguLubevXtMTExw9+5d4H6xPDs7\ni7W1NcPDw7S3t9Pd3U12djZ6vR4/Pz/kcjl1dXVotVqCgoIWdZimp6eprKxk06ZN6HQ6Ghsb2bRp\nE3V1dSK0dXR0lJycHDIyMmhsbGTt2rUcPnwYR0dH0ZH84IMPKC0tXZRHZiCKBwUF4ejoSG1tLXl5\neTQ3NzM8PMzk5CQzMzNoNBpUKhWNjY3iXn4aGIjLu3fvxtvbm6qqKvR6PTMzMxw5ckRkxFlYWHDn\nzh18fX1XHP75KNja2pKXl4eRkRFlZWV4eXl9o7l+/0hQq9UUFBSwffv2VUsWVjk8q3gMvLy88Pb2\npry8nF//+tf827/9G+3t7UxOTmJtbc3Vq1eRyWT4+flx+fJl4L60vaqqCo1Gw/79+0WCtp2dHSdO\nnKCjo4OSkhKysrIICQkhNjb2mUmQK8W2bdu4cOGCCB98Enx9ffnss88emZ7+oMuyn58fvr6+VFVV\nceXKFVQqlcg4Mqhc7OzsGBsbW9JON/CM7O3t+fDDDzl69CidnZ2sXbsWU1NTrly5wubNmzEyMkKn\n01FRUUFmZiY+Pj7PxNOB+0VOY2MjtbW1NDc34+bmxsLCAkeOHCEsLOypjqVQKHj55ZeFG7OB9/Sr\nX/1KXKc///nPrF27dkW7dEPHyIC2tjYcHR0xNTV9bLwJ3C/ADB2ut99+G1dXVyIjI3nppZeEeePj\nYGFhwdmzZ3nrrbf48MMPWb9+PWNjYzQ3N4vohvXr17NmzRp6e3tJT09HoVDw3HPPLUs6HxgY4OrV\nq0ilUkZGRnjxxReRSCSCyD87O8vs7Cy9vb309PTg6+tLaWmp6Bq1tLTw1ltvodPpRKilIYft4sWL\nyOVyamtrGRgYQKPRsHnzZjo6OmhsbESv1xMfHy8S3N9++208PDzESDYnJ2fZbu3s7Cx6vZ5f/OIX\nLCwsUFBQQH5+Pnq9npGREXp7exkfH8fS0lKMyfz8/HBwcMDe3v6xsu+FhQWKiopYt24dcrmcO3fu\nEBcXx6FDh7h69So+Pj5MTU1hb28vRlgrdT1/EoyMjLC3t2dkZIQTJ04I4YDBBPGfGQbC8mqxcx+r\nHZ5VPBbu7u4kJSXh4+ODhYUFNjY2qFQqrKysyM3NZWRkRJjghYeHo1QqcXV1xdfXl7CwMKKiorh3\n7x6tra2YmpoSGBhISEgI4eHhDA8Pk56eTn19PXK5HDs7u2/EGMvc3Jyenh7GxsaYnJzE2dn5sTJN\nMzMzCgoKhOv0w8jMzCQ+Pl609CUSCSqVinXr1qFWq0lJSUGj0VBVVYWnpycnT55EoVBw8+ZNYab3\nIFQqFR4eHly+fJn29naOHDnC0NCQMB3s7Ozk4sWLDAwMcPToUWJjY5+qPT0/P099fT1ZWVmkpKQw\nMzNDQECAWPRzcnI4cuTIM/l0KBQKAgICSE5OFlllBhgbG2NkZEReXh4RERGPXXT1ej0DAwN0dHQI\nro9CocDW1hY3NzeqqqrQarXLFqwymQylUsmvfvUrwsLCOHjwIDExMahUKmQy2RPvKb1eL3x32tvb\n0Wg0tLW1YWZmxqFDh9i5cyeBgYE4OjqiUCi4fv06ISEhVFVVsXv37kVdMJ1OR0FBAUlJSWzatIkD\nBw4wMTFBb28v69atw9bWFkdHR+FFc/PmTU6dOsWmTZsIDQ1l7dq1FBQUsG/fPqanp/nFL35BVFQU\nISEhKJVKRkdHRfSHgZ9jCOdsbm5GoVCgUCjo7OykubmZkZERFAoF09PTtLa2kp+fT1lZ2bJjQrlc\njru7OyqVCqVSiY+PD+vWraOnp4euri42b97M4OAgr7zyCjKZjHv37qHT6SgrKyMjI4OysjJaWlro\n6+tjcnJSFGsymYze3l4uXbpEeXm54FhFR0fT0dHB22+/TW5uLunp6bz22mviHsrJySEuLu5rGcVo\ntVqam5vZtGkT4eHhJCQkoNFoVqTG+y6jpaVFjKdXsdrhWcUTYGZmxo4dO0hOTl6kztDr9cKFuaWl\nBWNjY1pbWwkLC1ukclEoFLi5ueHo6Eh1dTUVFRUcPHgQJycntm7dypYtW6irq6OkpIT09HSioqJE\nFMjXia1bt/LBBx+siAD5oDz94cJofn6e+fn5ZXeecrmcuLg4tFotN2/eFDLs2dlZYmNj6erqIiUl\nZdnoAXd3d2JiYkRO0eDgIDExMVy5coXu7m527txJSEjIihfn+fl50clpamrC1dWVkJAQDhw4sKiI\nq6ysxMvL65n4GAZYWlryyiuv8OGHHwoFkQHr1q2jsrKSu3fvLop+MECn01FbW8ulS5f493//d6Kj\nozl06BB2dnbs2rWLpKQkfvKTn2BpaSmMC42MjNBqtYKUvH37dv77v//7qTpU09PTNDc3iz8mJiZC\nKRYfH49KpeKLL74gOzubw4cPC5Xh+Pg4bW1tbNmyBXNz80UP45GREdHV+dGPfiRGMdu2beMPf/gD\nUVFR4v7T6/UkJCQQERGx6PdleHgYGxsboqOjqaiooLKykqioKMzMzOjr68PW1pbg4GCys7OFd053\ndzfPPfcckZGRS4rWhYWFRR0ltVqNQqHgN7/5zZIxoUwmIzQ0lLy8PDIzM8W9ZihaUlNT0el0pKam\nis8+PT0tYjW0Wi1qtZqpqSnq6uq4ffs2w8PDImRZIpGIoGa4T87fs2cPU1NTaLVa5HI5p06doqKi\nAmNj46+twwMQGRnJjRs3KCsr48CBA/z4xz/ms88+Y3BwkCNHjnyl+/8fGauS9MVY7fCs4olwcnKi\nvLwcvV6/qN18+/ZtMfs3yI0jIyOXvP7evXvY29tz4MABJBIJV69eZXp6Gnd3dxQKBY6OjkRERODr\n60t7ezspKSn09vZiZmaGlZXV17IDUyqVDA0N0dfXh6+v7xONuBYWFqitrV3yEB0bG6O+vn5Zq/qF\nhQWuX79OZWUlRkZGvPHGG7S2tpKcnIxer2fLli3k5OQgl8uF+aABer2e5ORkDh06RFVVFb29vXR3\ndxMQEMCJEydwdnZ+4nWYn5+noaGBnJwckpKSmJycxN/fnwMHDrBu3TqcnZ2XLOy5ubn4+vo+sxLG\nAKVSiaenJ19++SXu7u5Cbi+RSHB1deXq1auEhYWJAsEQNnvp0iX6+/uJiYkReVoODg784Ac/wMXF\nha6uLubn50UitqenJw4ODmzcuJFDhw5x5MgR3nnnnSeqhhYWFujs7KS0tJSMjAyys7OF6Z6Br+Pn\n50dbWxvW1tZ4eXkRHh5OR0cHN2/exNPTE3NzcwoKCrCzs0OhUDA3N0dISAh6vZ6SkhIuX77MunXr\nliSlG2TiOTk5REZGIpFIKC4upquri2PHji3qQLW3tzM1NcXatWtxcXERRZFCoRD+NB0dHUISb2Ji\nwu7du4mIiFi2kyWVSsX7G+J1Ghsbsbe35969e8IcU6FQcPr0aWJiYoiLi2PPnj24urqKcZa9vT1B\nQUH09/czPj6OnZ0d7u7u1NbWYmxszPj4OAMDAwwODjI8PMzY2BhTU1MsLCyIRPmHPXXOnz9PVVWV\nKLwMBqUxMTF4enpSUFAgMtq+KqRSKf/1X//Fp59+io2NDTExMYSFhdHY2Eh+fj5+fn7/lKTeO3fu\n4Onp+Y3TBr4rWO3wrOKJkEgkHDhwgI8++ojAwEDMzMwYHx8XQZhGRkZs3bqVkpKSZV8/OzsrQi2j\noqIICAjg+vXr/PGPf2T//v2iI+Do6MiBAwfYuXMnFRUVJCcnC+O0Bx+Wz4rnnnuO3//+9yvKB1uz\nZg0JCQnMz88vKhIe5bI8NTXFl19+iUKh4MSJE1y7dg0LCwsOHDhAXFwcmZmZ/PnPfyY6OpqMjAyc\nnZ0XFT1NTU3IZDImJyfp6+vDwsICqVRKSEjIY3efBoWNwfDPxcWF4OBg9u7d+1ivILhfBDQ3Nz9z\nXMDDcHV15fjx41y8eJGzZ8+KRdaQqZWens7Ro0epqKggLy8Pa2trDhw4gJeXl0hKBwR3Sq/XExIS\nwqVLlyguLsbZ2ZnDhw/T0dFBT08PO3fuJDExkbm5uWXvjbGxMZqammhubqa1tRVbW1t8fHzYs2cP\nbm5uy47wHgwQlcvlHDhwgKqqKj7++GO2b99OWVkZZ8+epby8HJVKxfj4uBiPvPbaa48spCMiIigt\nLaW8vBwXFxdycnL44Q9/uOQchoaGxI7cycmJtWvXkpGRweHDhxkeHqaxsZHp6Wk2bNjAc889R11d\nHWVlZY9Ns3/wenz88ceEhoby+uuv89Zbb/HWW2+h0Wj4f//v/4muYENDA2lpaahUKvz9/YmPj2d0\ndFSQqa2trens7GRoaIjY2Fh6eno4e/bssgW5Tqdjbm6OCxcu0N3djYmJCW5ubnR2djI3N7fkNQae\nk1qtXnacvBJ0dHRgZmYmrBAA3n33XfLy8pibm+ONN97A39+fuLg4jh49Sn5+Pn/96185derUP51f\nzXLRHd9nrBY8q1gRVCoVERERXL9+nWPHjuHg4MCrr76Ki4sLv/vd7/D19SUzM5OZmZklC9XDOVpm\nZmYcO3aMlpYWkpOTKS8vZ+/evaKQMDY2JjY2VniZlJSUkJmZSWhoKDExMc8sU7W2tsbCwmLZzs3D\nMPBR2tvbF8mql3NZ7urq4osvviA8PJytW7cyMDCw6EFmZ2fHyZMn6e7uJiMjA7lczvnz5/n5z38u\nrtWtW7eYm5sjPz8fOzs7fvazn1FWVsbf/vY3jh8/LhLM4e9FTm1tLY2NjTg5ORESEsKePXuW5Eg9\nDh0dHdja2n6tihVDQXHhwgXOnTsnvD82bNjAH/7wB95++21cXFw4duzYkqBPA0ZHR6moqKCqqgqZ\nTCZIwYZsvLi4OFpbW0lJSREuuuvWrWN+fp62tjZR5MzOzuLj4yPk9U8qAGH5xPTQ0FCcnJz4+OOP\ngfv3UV9fHw4ODvzlL38hLi6OTZs2PZYrJJFI2L9/P5988gkmJibs2rVr2ZiQ4eHhRd/1tm3b+P3v\nf8/nn39OQ0MDEomEH//4x6JYDg4O5vr160+MDxgaGuLjjz9mw4YNojtpa2uLra2tcHo2MzMjIiKC\niIgItFotra2tNDQ08Omnnwqulq+vLxYWFqID2tfXh0KhIC8vTwSDPgipVIqpqSkHDhxALpdjb2/P\n/Pw8//M///PIINeNGzcyMzPzzOOsTz75RJCyraysBCfvwY1OU1OTMPbcvHkz9vb2fPLJJ+zdu/ex\nCe7fJeh0OkZGRp4YR/N9wupIaxUrhru7OxkZGahUKiHBlslkjIyMMD09jV6vR6lULilIiouLCQwM\nXOIqbOAqDA0NkZiYiFwux8XFRezKJBIJNjY2hISEEBYWxuDgIGlpaTQ2NmJkZIStre1Tk5zr6+vp\n6OggPDz8iR2jyclJ+vv7FxU8TU1NguMDUFpaSmJiIgcOHCAmJgaJRML4+DjNzc1LOCuGVHZHR0fq\n6uooKirCysqK69ev09XVxZ49e+jo6GDfvn3Y2tqK2IVLly6JYNbc3FySkpIYGxvD19eX/fv3Exsb\ni4uLy1OHIxYVFeHk5LRiZ+GVQqVSIZVKSUtLw8/Pj7KyMq5cuSJM9V577bUli7BaraaiooLU1FRu\n376NnZ0d27ZtY/v27SIM1tbWVnQ/bGxsiIqKorGxUXROsrOzRaTF5s2b2b17N0FBQahUqhVfm/7+\nfiYnJxclpsP9Ir2mpgYzMzMKCwvp7e1ldnaWl1566YlZZQZYWFhQXl6OTqfj8OHDy74mLy+P8PBw\noYTKycmhq6uLsbExMRZ9sJsjlUpRq9V0dnYuOWcDenp6OH/+PNu3b19k/jk4OLhoLG2wFjAc187O\nTnRCPDw8GBsbo7Gxka6uLhwcHNiwYQO+vr50dXVRX19PWVmZME18uLi0sLDAzMwMiURCeno6VlZW\njI2N8fOf/5zs7Gymp6fx8PAgKSkJf39/BgcH6enpWRTbsRJotVrq6+uZmJhAr9ejVquZnJxEKpVS\nVVWFkZERJiYmvPPOO4s2Zvb29vj6+pKYmMjMzIzoOn6XMTY2Rk1NzRPT5r9PWC14VrFiyGQybGxs\nSEtLIyoqShQbRkZGFBYW4uvry+Dg4BKjuYKCAiIiIpbdYUulUry9vfH396egoEAY4D3cdTCkNq9f\nvx6FQkFxcTFZWVloNBrs7OxWPO6qqanBwcHhsT47BigUCvLz84mNjV30eisrK5ydnUlKSqK2tpZX\nXnllUbdibGyMtra2ZflMEokEW1tbIiMjKSkpoaqqShA/zc3NGR8fZ+vWrUgkErRaLUNDQ0xPT1Nc\nXEx7ezshISHs37+f9evX4+rq+swJ0Hq9ntTUVOLj478RTxIHBwcaGhrIyMhAoVBw6NAh4uPj6e/v\np7u7W+QkNTQ0kJmZSVpaGlKplNjYWDHmNEj/pVIptra2pKWlERwcTFNTE7dv3yYtLQ21Wi2uqV6v\nZ/v27URFRYnXPi2Gh4cZGBhYomrp6+ujtLSUjRs3UlVVxcLCAnv37l3UjXkS6uvruXfvHvPz8/j4\n+Czpxun1eq5fv87GjRspKCjg6tWrODg4EB0dzb1790Rq+8MbCjs7O1JSUoiJiVnSMWlra+Ozzz7j\n4MGDS8Ze169fF9Edvb29hIeHLxsBY8jd8vLywsbGhunpaZGDZchDUygUjI+PC9+hsrIypqamMDEx\nwdzcXHwXbW1t5Obm8tJLL1FVVYW5uTkBAQG8++67/Od//qcYKXV3dzM+Pr6i/KvR0VGqqqrIzs4m\nJSUFqVTK3Nwcer0emUyGg4MDP/3pT/nlL39JdHQ0YWFh7NixY8lxzM3NCQ0NpaCgQEStfJcTxru7\nuxkeHiY8PPz/+lT+YbBa8KziqWBvb09LSwujo6OiM2BlZUV2djZr164V44UHsRJ5qZmZGeHh4SKD\nanJyEg8PjyULjsEQMTIykjVr1ojRRl9f34pIzpWVlWJRCw4Ofmzb3NzcnNzcXIKCgsTPlZWVYWtr\nS3p6unDMfZjTMzIyQmdn57K7U71eT0VFBV988QXu7u5MTk6ysLDA6OgoDQ0NbNu2jfHxcXJzc0lM\nTGRkZISAgAB27dpFb28vo6OjrF279iubNg4NDVFRUcHOnTu/1p2sWq0mLy+Py5cv4+zsLAJl4+Li\nhFllUlIS3d3dpKWlMTo6SlBQEIcOHSIsLGyJNYFBhdTe3k5bWxsFBQVoNBo8PDzYvn0727Zto7Cw\nkDNnzuDq6kpaWhqtra24ubk9U37bxMQE7e3tS0aeN27cEIThjRs3Mjc3R319PWNjY3h7ez+x0zg5\nOcknn3zCyZMnUalUy0r1h4eHRbfKysqKEydOYG5uTmJiIocOHaK8vJygoKAlBY+JiYkgdz9IPm9o\naODy5cs8//zzS7o/arWaq1evLpL5z8zMPDY8Fu7zvsrKyjh69ChhYWHExsZiYmKCWq1mYGBAOLL7\n+voikUjIycmhuLiY8fFxpFIpiYmJ7N+/HycnJ0ZGRgRp2c7OblGnsb29nbm5uWU3JYZxW3FxMWlp\naZSUlKBQKAgKCmL//v2EhYVRUFAgxlWG62hhYUFgYCD5+fmEhoYuux4ZGRkRGhpKa2srubm5+Pn5\nfes5gF8XGhoaAJ64sfs+YbXgWcVTw93dnYSEBFEIGMY4BtJjTEyMeCDr9Xpu3brFtm3bnvhQMKQ1\nh4eHU19fz/Xr17GxsXkkN8Hc3Bx/f3/hf3Pr1i3u3r2LRCLB3t5+2d2ZQYZtYWFBXV3dY/0pJBIJ\n/f39aLVasfPMzs6moaGByMhI9u7du2zhMTw8TE9Pz5KHZkdHB1988QW9vb0cOXKEjRs3CmWLj48P\nY2NjVFVVCd7QwYMHiYuLw9XVFTMzM0JDQ+nq6iIrKwtfX9+vJNktLy9HqVQ+k1vucjCMXxISErCw\nsODYsWNERUURFBREfX09VVVVjIyMkJ6ejk6nY3Jyktdee40NGzbg7Oy86DpOTExQW1u7KJ/K2tqa\n8PBw2traePnll/Hx8UGpVCKRSISzd0hICNHR0YyNjZGQkMDCwgKurq5PNfacmZmhrq6OqKgo8W9V\nVVVkZWURFBTEqVOnaGtrE5+xoqKCkpISERC6HPR6PRcvXiQwMJCIiAicnJwoLS1FLpfj5OTE/Pw8\nhYWFXL16Fblczg9/+EMiIyMZHBzk4sWLnDx5En9/f4qLi5menl6Wf6ZUKsnJyWHdunVIJBIqKytJ\nSUnh5ZdfXpYrZVBPGbx99u/fT2Bg4BO7fUZGRmRmZrJlyxaR+u7g4EBwcDCOjo60tbXh4+NDbW0t\nPT09eHt7ExISgkajISMjQxDMjYyM6O/vp6enB7hviGlQsAE0NzcjlUpFB210dJTq6mrRxRkcHMTJ\nyYlNmzaJ0aWjoyOTk5PCEPX06dMix84AiURCT08PUql0kWfUg5BKpQQEBKDVaklISFikOvwuoaKi\nAnt7e9zc3P6vT+UfBqsFzyqeGgbFVXFxMaGhoUgkEkxMTMjPz8fe3h4LCwvBtdBqtU8tLzUyMiIw\nMBAnJyfS09NpbW3F3d39kQ8UuVyOm5sbMTEx2NnZUV1dTVpaGpOTk9jY2Cya1VdUVODu7k5ISIjI\nqXocmdXACQgMDKSoqIjq6mrBm3lUZ2RwcJD+/n5BfhwbGyM5OZnCwkK2bNnC3r17MTc3p6Ghgezs\nbOC+M69hnDM/P8/du3dRKBQ4OzuLwk0ikeDv749er+fq1au4uro+cyBgRkYGUVFRX9mjY3Jykqys\nLK5du4adnR3Hjh0jIiICU1NTZmdnqaqqorOzk/b2dqanpzl8+DB79uyhqakJvV6Pu7s7Wq1WkNOv\nX79OQUEBcJ8AvXv3bjZv3oyPjw9OTq0pDx8AACAASURBVE5oNBqqq6sXueTOzc3R0tIigkw9PT0J\nCQkRmVf29vYrJm7Oz89TVlbG+vXrmZ2dJSkpiaKiItzd3Tl58iQymYzi4mK8vb2Ft5HBUdvR0XHZ\n97l9+zb9/f0cPXoUiUSCRCLBycmJa9euodfruXTpkvhuLSwsiIyMpLe3l08//ZRjx46Jh/7t27dR\nq9WoVKolcQvW1taCk2W4r86cObPE/sAAuVzOmjVrUCgU1NbW8uKLL65otCmTyUS0ysO/jw4ODnR0\ndODg4MArr7yCVqulsrKS/v5+Wltb0ev1giicnZ1NW1ubeO3U1BRubm7ifqypqUGj0dDa2rpsFyc2\nNhZPT08xutTr9RQVFXHlyhWio6NFgOpymJ2dpa2t7YmbHXd3dxwcHPjyyy8xNzd/ZIH0j4rCwkIC\nAgJWScsPYLXgWcUzwcXFhdu3b2NqaoqjoyOWlpbk5+eLMY2BAGnI/9m4ceNTv4eBmDoyMsK1a9eQ\nyWSLSM0Pw0ByXrt2LaGhofT395OWlkZTUxNGRkbY2dlRVVWFh4eHGAtUVlY+lidgYWFBWloaubm5\nIu16dHSU/v7+R7aKBwYGGBoaws/Pj+zsbBITE/Hz8+Po0aPMzs6Sl5dHYmIibW1tmJubExISwuTk\nJHZ2dkRGRuLv709gYCDV1dXcuHEDhUIhiMBwX/6tUqn48ssvMTMze+qFeGZmhszMTA4cOPDMztZj\nY2PcvHmT5ORkVCoVx48fJzQ0FIVCQVNTE7du3SIlJQW9Xs+6devYu3cvlZWVTExMYGNjIzhQLS0t\npKenMzg4iKOjo/CACQkJwdnZecnYwc3NjZs3b+Lk5CSKPaVSyc2bN9m4caO4N0xMTFi7di02Njak\npKTQ0dGx4jFXQUEBTk5OXLhwAQcHByYnJ9m7d694gGZkZLBx40bRYfLw8BBeQ7Ozs4vce3t7e0lO\nTub06dOi8DbYAdTX1zM0NMSpU6eIi4ujvr4eGxsbjI2NRfyBoQOn1+vJzMzk0KFDXL9+fRGHDu7f\n+zqdjtzcXDo6Onj11VdXJEdWq9WUlZXx3HPPPfFnDaivr1+26ALw8PDg2rVrBAYGsnbtWsLCwuju\n7mZ0dBR/f39mZ2epqalBKpUuSXA3xFeUlJRQW1sr8u8eJKAb3K4fxPDwMJ999hlDQ0O89NJL+Pv7\nP3ZMa7hfNmzY8MRxroG4nZSUJMJZvytk5szMTDZt2vSdHcl9E1gteFbxTDC0hBMTE0Vy8uTkpAgP\nNYwExsfHaWhoWET8fdr38fLyIiAggNu3b1NSUoKzs/MTd6MmJiasWbOG2NhYsSvPzs5mbm4Od3d3\nHB0dcXZ25saNG3h7ez/yeFNTUxQXF2NiYiIM0gzS+0dJ23t7e2lpaSE3NxdjY2PWr19Pf38/KSkp\nDAwM4OXlxd69e6murmbnzp3k5OTwyiuvkJubi1KpFNb+wcHBeHp6UlRURH5+vkhwN5B0/f39SU5O\nZmJigjVr1qx4Ia6trWVhYeGps7Pg/kPpxo0bpKen4+7uzokTJwgKCmJkZIS8vDyuXbvG4OAgAQEB\nHDp0iIiICJGz5O/vT2xsLO+88w7u7u4iaPX1119n/fr1eHl5YWlp+djPIZPJsLKyEh0qqVSKiYkJ\nZWVleHh4LPke7ezsiI6OFkUz8Ngx18LCAllZWbS2tnLkyBEcHBxob28X4Yuzs7Pk5uaye/fuRedp\nbW1NaGgohYWFVFVVCeL++fPn2blzJ56enuh0Oqqqqvjyyy8ZHx9nz5491NbWCsK6oUOTmJjIrl27\nFpGMNRoNhYWFHD9+nJaWFsbGxhZleOn1eiorK2lqauLMmTMrtm4wZFw9TQe2vb0dhUKxrFmlsbEx\ncrmcgoICQYLu7e1FIpEwMDCAo6MjL7zwAi4uLrS1tS0qeqanpxkdHWVkZARTU1NCQ0PZsmUL1tbW\nj/T4uX37NgkJCcTExCxxEX8UTExMKC0tFWaST4JhnFxUVERdXR3+/v7fSvDxV4FGoyE7O/tr5+h9\n17Fa8KzimWFlZcXQ0BDt7e34+flhampKYWEhIyMjrF+/HrlczvDwMB0dHYs4Ec8CpVJJeHg4CoWC\nhIQExsfH8fDweOLCI5VKUalUjI6O0tfXJ/ghhYWFODg4CD7Fcg//pqYmPv30U9FqHxsbQ6fToVAo\nOHv27LKkx87OTpE+7ubmRnt7OyMjI3h6erJ37142btyIu7s7HR0dwmjP3t6eqKgoPD09uXz5Mn5+\nfmLMZmFhQVhYGPb29mRlZVFeXo6trS3W1tZiIS4sLKS+vn7FqhIDGfNR447lMDg4yPXr18nIyMDb\n21uktJeXl5OUlERNTQ1ubm7iM7q4uCzZiRsbG1NRUYG/vz9vv/22UKpZWVk9lROsvb09dXV1zM7O\nCn7C8PCwkDY/DJlMhpeXF0FBQdy5c4e8vDwcHByWdCja29v55JNP0Gg0/PSnP8XFxYXr168TGhoq\nHu5dXV0MDg4uez8bGRkRFhbGyMgIKSkpdHZ2YmtrS3x8PHV1dXz55ZciTmHr1q3Y2dlhZGQkioOb\nN2/S2trK1q1blyj8pqamqK6uZsOGDbi5uZGQkCBI9zqdjsTERAYGBvD29hay6pVAJpORl5f3VAXP\n4OAgU1NTj1Soubi4cOfOHaRSKVqtllu3bvGDH/yA2NhYGhoaSE1Npa+vj/HxcUGalslkqFQq1Go1\n3t7ejI+PMz09za1bt+jp6UGn02FtbS1+3wcHB/n8888ZGxvj5ZdfFiTplWJ4eJiZmZlHekE9DIVC\nQWhoKO3t7WRnZ39lDt03jYGBAdrb2595o/nPitWCZxVfCR4eHqSmpuLp6YmLiwvFxcVYWlpiZ2eH\nra0t/f39DA4Ofi1mXgbuQ0REBI2NjVy/fh1ra2vR9XgcRkZGRNih4Vjd3d1i/OTk5CRm3Xq9ntzc\nXDIzMzl58qRIjNdoNCwsLLBv3z68vb0XHX90dJSLFy+Sl5eHRqMRC+SDRc6DreXk5GRCQkIoKiri\n+eefx9jYGHNzc5RKJWlpaYSHh4vF3dDRiY6OxsjIiLS0NJqbm3F0dMTGxmZFqhKdTsfQ0BDGxsYk\nJyezb9++FUna+/v7SU1NJSsrS8RUzM7OkpmZSXZ2NpaWlsTHx7Nr1y68vb2f+BB44YUXeOGFF4Tc\n3MBliYyMXPGu2RBXkZCQQHh4uEiTr66ufqwE19A1sLCwIDk5me7ubtzd3ZFKpdy8eZOsrCz27dtH\nV1cXYWFhaDQaMjMzFwWr1tXVIZVKHznOlEgkeHt7o1arhRtzTk4O7e3t7Ny5kx07diwqtJydnSku\nLmZ+fp7a2lqee+451q9fv+S4o6OjNDc3s27dOkxMTNDpdJSWlhIYGMjly5eZmZnh5Zdfxt7envT0\ndGJjY1c0rpTJZGRnZ7Nx48YVS7Cnpqbo6Oh45ChYIpHg4uLC1atXaWxsJCIigtbWVm7evEl/fz+u\nrq6Mj49jYWGBvb09MzMzyOVyfvUv/0KEnzUmC/0MdNYyOTOPg9P9MWR7ezs3btygs7OTuro6bt26\nRVxcHPv27XumwmMl98vDMHzvOp3uK3PovmkYOHMrkfV/n7Ba8KziK0GhUGBmZkZmZiZRUVHMzMww\nPj6ORCJhzZo1dHV1MTk5+bWm9SoUCgIDA3FxcSE9PZ3m5ubHkprhvhleSUkJ8/PzyGQydu3axfHj\nx7GxsaGrq0sQSw2eJn19fZw5cwaVSoWFhQVZWVm4urqi0Wg4ceKE4Ew0NjZy6dIlbt68iVarZcOG\nDWJcs2PHjmXPqa+vj6KiIoyMjPD09Fx0bZydnenr66O2tnaJoZ0hlT0mJoa5uTkSExPp7+/HxcWF\n8PBw5ufnSUxMxMPDY4lUvq+vj//93/+lsLBQcKGsra0f+VDs6ekhOTmZ/Px8oS5qa2sjPT0drVZL\nVFQUBw8eFIaSz9o2t7KyYnh4mJaWlqeSzyqVSqanp2loaBDqouvXrxMXF/fYB7dEIsHBwYGoqCgG\nBga4evUqJSUlGBsbc/r0aZydnamsrMTb25vKykocHBwWKdnu3LmDm5vbY7tjExMTJCQkYGVlRVdX\nF+bm5uJeWi5KwcrKioSEBExNTXnxxReXPebDRnxubm7k5uZSXl6OmZkZJ0+eRKFQYG5uTl1dHSYm\nJisea2VnZxMZGbniwkGr1VJeXr7EfsIAgw9Vc3OzyNNydHRk06ZN7Nmzh/DwcGJjY6mtrWWgt529\nAcNsUnyMRcP/h1HbX3GYSmGtaRFbbHNYa5KPpb6TtkEpOiN7kdOl1WrFHysrq6f2o7KwsCA9Pf2J\n98vDkEgkuLm5oVKpuHTpEiYmJk/VKf22UFtbK8b6q/g7VgueVXxlqFQqoaoICAigsLCQiYkJhoaG\naGxsRCqV4uDggImJyTOTZJeDtbU1UVFRIs/IsPNf7uErlUpFcKJCoeD5559HJpNha2uLjY2NkE3f\nuXOHubk5tm7dKo41NjbGu+++yxdffMHU1BTm5uY0Nzdz5coVKisrUSqVvPDCC+zbtw9PT096e3vR\naDRLDBgNyMjIwNXVldraWp5//vklox8fHx9u376NVqtdVlIqlUpxdXUVLtXXrl1jamqK9evXCzLz\nw2Miw+jE4JNSXV2NUqlccvzOzk6hTPLy8sLOzo7y8nIGBgbw9/fn4MGDREVF4eDg8LV9lx4eHqSl\npeHq6vpU8l83NzfBJbK1taWpqQkrK6sVq1La29vp7e3F3NwctVotduy1tbU4ODiQnZ29hBeSlZVF\ndHT0IzlfHR0dfPDBB0gkEnbs2MHRo0cZGhrixo0by3KMZmdnxb1rZGS0bHcHlhrxzc3NUV5eztjY\nGD/4wQ8WPfAVCgV37txZsUtxTk4OISEhK772CoWCzMxMNm/eLAwy29raKC4uJj09naKiIjQaDePj\n45iamrJv3z4iIyMXmUFKJRKmKt/hlMN7OMzfwUo+DHot6OdBN4dcsoAEHTLdNFb6DsLM7uAiu8e8\n7RbmdCbo9XpMTU3p7e0lMzOT5uZm5ubmsLS0XJEJqVwup6mpCWtr62dSMRk4dCkpKYyOjj4Vh+7b\nQGlpKW5ubt85Zdk3jdWCZxVfGYZdz5UrV+jp6WFoaIiZmRmxSI+NjXHnzh3Mzc2/9nA+gww5KCiI\noqIiioqKcHZ2XjbgU6VSkZ+fz7p16xZ1Ey5fvszk5CR6vR4TExPi4+MpLS0lNzeX6elpDh06RGlp\nqSjgLl26hI2NDW5ubpw8eZIdO3Yseli0t7ezsLCwyKrfgMnJSdLS0jAxMSEoKGjZnzH4jyQkJODm\n5vbIB5FcLhep3m1tbSQlJWFjY0N8fDxJSUmLFENyuZyioiK0Wq3Ip9q3bx9SqRS9Xk9bWxvXrl2j\ntLQUa2tr5ufn6e3txdXVlT179rB58+av5Oz8OMjlchGx8bD66EmvMzMz49atW0RGRjI5OcnQ0NCy\n1/RB9Pf3c+HCBbRaLadPnxaKK0PHzJDuLZPJRO4U3O9s3Lhxgz179iw5x97eXhITE8nLy8PMzIxf\n/vKXuLi4IJPJ8PPzw9zcfFFHQCKRoNFouHDhAi4uLri4uNDS0kJQUNCyNgnt7e3Mzs4SEBDA5OQk\nH330kci1GhgYWLSTt7OzIzMzE19f3xXlh+Xm5uLv77/ikEm5XM7t27cByM/PJzU1VRCSN27cyI4d\nOygoKGD37t0EBweTmppKdHT036/Z/AT6zL04z6SgYA6ZRPvY95OgR8oCVopJguTZxG5/mcC4+13W\nkZERJicnMTExYXBwkOzsbOrr65mdncXc3PyxXauJiQnhgfUsMAgX7ty5Q01NzT8UmfnBiJJV/B2r\nBc8qvhZMTk5SWVnJ0NDQkv8zFBJHjx79xqzaTU1NRaJ6QkICY2NjuLu7iwWot7eX3/3ud+Tk5NDc\n3IyVlRVeXl50d3eTm5sryJNarZa5uTnOnTuHkZER77//Pvn5+UKhBffn/1qtlr/85S/LzvBbW1sB\nlvB84P4DwtTUlM7OTo4fP/7I62Fqaoq9vT1Xr14lNDT0sYWGkZERfn5+BAUFUVNTQ35+PuvXr6eh\noYHW1lb8/PyQyWTU1NQwNTWFs7MzZ86cQSaTiU5VeXm5yB4yZFHt2bOHNWvWPHNq9dPA3t6e5ubm\nJeqjJ8HR0ZGqqirh6VNYWEhMTMyyP6vT6cjLyyM5OZn4+Hh27tyJsbExEokER0dHoqKi6Onpobq6\nmuHhYbZt27ZoLNTf3097e/uiLszAwIAY/Xl7e9Pf388Pf/jDJdfM0dGRgIAAbty4QVdXF56enly8\neBEbGxsOHjxIZWUldnZ2Itj24W5Bc3OziHb58MMPCQ8PZ9u2bcK5+kE/KUO0Qnt7+4rGhHl5eXh5\neT12NPNwF2dqagqJREJYWBj79u1bpLLLyspCIpGwdetW7O3t6ejoYHBw8P7vw/wEpG+AsUrkkvlH\nvt9ykKAD3Tz0pGBq64Nb2H0V4Lp168SIc3R0FIlEwvDwMPn5+VRXV6NWqzEzM1vyncjl8sfeLyuB\ngavX3d0tisz/azKzXq/nxo0bbN++fUn3+PuO1YJnFV8Z8/PzvPPOO0t8NQyQyWQcPHhwWRnr1wkD\nxyUyMpLm5mbS09OxtLTk1q1bbN++nby8PBobG6mqquLKlSukpqYikUhQq9WYmJiwsLAA3B8zlJSU\nMDg4SG9vL3fv3l1kwW/Ar3/962XPo6WlRSiDHsT8/DxXrlxBoVAQHR39xAe7nZ0darWawsLCZR+C\nD8PU1JTg4GC8vLy4e/cuU1NTwP32tr+/Py0tLczOzvL666/T3NzMxYsXKS8vF6Zv69ev59ChQwQH\nBz9SCvxNweBnY/BwWWmRZXDnvnbtGlu2bCE3N5eIiIglBeLQ0BCffvopk5OTnD59etlwSLlcjo+P\nD729vfT39wsyu2GX3NjYyMLCgpDhp6amcuvWLUJCQjh48CAZGRns2bMHd3f3Zc9VqVQSERFBfX09\nycnJ2NjYcOLECaRSKfn5+WzZsoV79+6hVCpxdHRkbm5OFOx1dXUia2vTpk3CQ8YgA799+zbh4eHi\nMz2Yr/WkrkNBQQEuLi5LzntsbIzq6mpycnKEpYKhi2MYXxuUkwb09PRw48YNXnrpJTFa8vDwIDEx\nEX8/P8zunILRciT65deKFUE/D33XwWkHKN2ET1VwcDAbNmxApVKh1WoZHx9ndnaWsbExCgsLqaio\nYHp6GqVSiZmZGRYWFty6dWtFQcKPg0Qiwc/PD6lUypUrV3B2dl7Wo+jbwtTUFHfv3n0qb6XvC1YL\nnlV8ZchkMnx8fISvxsPFgaWl5SPTob8JKBQKAgICcHV15aOPPuLXv/41c3NzaLV/b51rNBoGBwdR\nKpWcPHmS5uZmcd46nY6TJ0+ye/duTE1NuXTp0qIOj0QiYcOGDZw5c2bZ9zcYHT4seTUEKk5MTHDk\n/2fvvYPqyu887edG4JJzvAQRRY4SyiiHllooSy3J3e5u99gz9sy7W95al7dm9o+dqvHWeD1jT3nH\nobzuqFa3WiAEAhGEAAmQEDlnEDlccrhwueH9Q3PPiCYIqdVt2c1TRal0w+Gccw/3fH/f8PkcP76m\n0o2Xlxe1tbWoVKo1NyBaWloSERGBg4MDra2tTE1N8fd///ekpKQQGhpKRUUFNTU1mJiYsHXrVl5/\n/XViY2NxcnL6k5olmpqaIpFIlty8n4WFhYXQKKtQKJBKpTg7OwMICrwpKSnCVM+zhNiKi4uxtrYm\nPj6emzdvolKpUCqV1NbWYmlpSU1NDVlZWfj7+3Pq1Cl8fHzIyMjA2tqaHTt2rLptkUhEbW0t8GQ0\n2sbGBkdHR2FF7ubmxs2bN+nv7yc5OZnY2FjkcrmgAWPsh3ka43SkXC4XjtvExERwdH9WGdko0aBU\nKgXF69u3b/Pw4UOkUilBQUFLsjiDg4PCCLkRrVbLlStX2LNnz6LgycTEBJlMRn/xP+OlyUCkV6+6\nP2vCoIX+LPD/AYj/M6ATi8XY2tri5+fH5s2bhezm7OwsY2NjQnm9rKyMmZkZDAYDJiYmL6XXxc3N\nDVdXV65fv45MJnvp5fu10tfXx9DQ0LLmxd921gOedV4KVlZWxMXFIZfLaW9vx8zMjPn5eSwtLdm9\ne/efZJLB2tqajz76SCjXfBmtVktbWxsbN25Er9djMBiQSqXo9XocHBzw8vLCx8eHwsJC+vv7WVhY\nwNTUFHNzc06cOMH09DTe3t5LVtCtra2YmZkt+tI3GAwkJyej1+vZvn37mrNdIpEIPz8/MjIyVvUV\nWw47Ozuio6Pp7e3lV7/6FaOjoxQWFvL2229z6tQp9u7di1Kp/Fr6cl4UNzc3Hj16JPhMrRUPDw/S\n09Px9fVlcHCQoKAgxsbG+OyzzxgaGlqzVotarebOnTu4ubkJzutdXV2kpaUxMDBAd3c3Pj4+nD59\nGj8/P6RSKbW1tVRVVXH+/PlVA0aDwcDNmzeZnZ3lzTffxM/Pj1u3bjE4OMjIyAi7du3i7t279Pb2\nMjw8jFQqJTQ0lKGhIQoKCtiyZcuiniIjT1tVREdHC9ejubk5eXl5xMXFrXjcxgBApVJRUFDA4OAg\njo6OwkTVxo0bcXZ2XlIamZycpLe3d5HFR35+PjqdThBpfBpXJ3vcGi4hNcyuev6fC70WpFbguGXZ\np0UiEebm5nh5eREdHU1UVBTm5uYsLCwwNDTE6Ogoo6OjNDc3o1arkcvlzxS+fBa2trYEBQWRmZkp\nLFJe5qDGWmhra0On0700n7y/JF6NDqt1/iIQi8Vs2bIFd3d3fvSjH5Geno6ZmRl///d/z+XLl/m3\nf/u3r5Q6fhHy8vKEUtVymJiYEBwczKFDh9BqtYyPj3Pt2rX/nCb5D4fn5ORk/vf//t/Ex8fz/e9/\nHz8/PzIzM/nNb37D8ePHF5WvdDrdkhtfS0sLBoPhhRSOjWPHn3766Yp+TSshEonYuXOnENQ4ODjg\n5ub2XEJ/3yRisZhjx47xySefLBJgfBZmZmbs2bOH0tJSpqamKC0t5e7du2zbtk1wal8LFRUVuLm5\nCRk9vV6PVCpFp9Oh0WhwdHQUTHPhScCQkZHBxYsXVw0cDQaDMNFz8eJFpFIprq6uvPfee3z66afo\n9XrS09OpqakR3iMSiWhsbKSkpAR7e/sVp/7gScAXEBDA3bt3OXz4MIDQsN7Z2SlkYrRaLV1dXbS2\nttLa2srMzAx6vR4rKyvee++9NZcSbW1tGR0dFf7f399PaWkp3//+95efkuy7iUwqhpX/FJ8f3Sw0\n/hyC/j9YQ5BiYWFBdHQ00dHRLCws0NHRQXV1NXV1dVRVVVFdXY3BYCA4OJiQkBC8vLxeKFixs7Pj\nnXfe4fr163zyySecOXPmG+3rUalUX9kj7y+V9QzPOi+d7373u2RmZqLRaNBoNGi1WhoaGmhoaOD0\n6dPf6L78+te/XvTF/GWM+is+Pj6IxWIUCgUuLi7k5OQQFxeHWCwWJsFmZ2exsrKisbGRmJgYwsPD\nsbe3F5Sfvby8kEgkNDY2Ym1tvSiLk5aWxuzsLPv373+hYMPKygqZTEZOTg4RERHPVXqysLDgzJkz\nDA4OcurUKSYnJ+np6cHPz+8bX32uBUtLS6anp5/pZv9lXFxcKC0tZWJigtHRUd544w2CgoLWvGI3\nmrLGxMTQ1dXF5OQkN27cwM7Ojn379tHZ2cmOHTtISUlhdHQUd3d3rl+/TmRk5KoCbwaDgZycHLq7\nu7l06dKioN8YTKlUKtra2oTHRSIROp2O/v5+vvOd71BfX09kZOSqAaBSqSQ9PV2wSjEet1E009iL\nY8zibN26VbC3MAYDa0UqlZKfn8/27dvR6XRcuXKFhISElZWLy/4W0VTzmre/dgzgegAUz9cfKJFI\nsLe3Jzg4mOrqavbt24eNjQ3j4+N0dXXR2NjIvXv3UKlUSKXSVTWrlsOYnRscHCQ7OxtfX99vpPkf\noKSkBD8/v+fKBn9bePW+7db5s6ajo4OMjAzm5uYWPa5Wq7lx4wY9PT2LHquqqhKmmr4Ozp07t2pW\nSa/Xs2XL4pS4UqkULCee3lcTExMhiPvDH/7AzMwM/v7+/OAHP2BmZobf/va39PT0LMnwGJtgFQoF\nQUFBL3wscXFxODo6Cqacz4O/vz8xMTHMzMywY8cONBoNH330ETMzMy+8P18nCQkJdHV1LQoCVsNg\nMFBTU8P4+DgAYWFhzx1YtrW1IZfLhWb1iYkJvve973H06FGmp6cFle+/+Zu/QSQS8ctf/pLZ2dkl\n18+XKSgooLW1dUmwY2RwcJDJyUnh/2KxWJAi2LhxI9PT0zQ1NS3qI1sOhULB3r17BVPWrKwsSkpK\naG1tpa2tjeDgYP72b/+Wd999d5HOlFwuX3HgYLXfZZzqu3//PlZWVoSHh6PT6RYdi8Bo+XNtf60Y\nDDoYLf1K2/D19WV6epo9e/bwwx/+kL/+679m165dgr5YUlISP/vZz/jss89obGx85udgRCwWC5IO\nf/zjHwXz4a+b9QzPyqxneNZ5qRQUFHDz5k3m5+eXPGdhYUFUVBRjY2NkZmaSnZ1NU1OTMFb9dRAc\nHMxvfvObZfdHoVDw3/7bf2Pfvn1LnnN0dOTWrVvExsYikUgYHR2lvb0dtVqNXq9nYWGBhoYGwsPD\nhekoc3NzwTHb09NT6EHJzs5mbGyMo0ePvpDImRFjP09+fr5QEnkeiouL0Wg0NDY2cvToUbRaLbdv\n32bDhg1rLh19UxhX4Onp6URHR6+a0ZqZmSE5OZmmpibOnz+PSqWis7Nz2X6XldBqtXz++eeMj49j\nZmbG9PQ07777rlCKqK6uFqQMZDIZZmZm1NfXo1AoqKmpwdXVdVkjyqKiIqqqqnjzzTdXPMcPHjwQ\neseMP3Nzc0RERPA//sf/4J/+6Z8oKyvjV7/6FcPDw+zfv39JtmFiYoLa2lpaWlp4/Pgxjx8/xt3d\nnW3btiEWi7GxsSE+Pn7ZMeX6d8xF+gAAIABJREFU+nrm5uaIiYlZ8/kSiUTU1dVhbm5Ofn4+b7zx\nBqamprS1tfH73/9e8BGztraG+RGo/xkYXmY96z/2w6BFNSXCLODCC/fe6HQ6QQ4AnjTPe3h4EBUV\nxaZNm3B0dBT6/YyZn8ePHyORSLCxsXlmttXV1VXQKTOKhn5dAxxarZY7d+4se42ssx7wrPOSmZyc\n5P333192FSSVSomIiKC5uVkQ+oMnJQxjo6WJiclL/UO1sLDg8OHDpKamAv+pCSSRSPje976HhYUF\no6OjWFpaLioDWFhY0NfXJ+jCDA0N0dnZyezsLHq9HrFYjF6vx9/fXxhbdnJyIjw8nAcPHtDR0YFe\nr8fa2pqMjAycnJzYvXv3V/6ik0gk+Pj4kJSUtKrL+5fR6/XcuXNHOAe1tbUcPnwYR0dHrl+/jpOT\n0yu3KrS3t6e7u5vBwcEVJ9QaGhr49NNP8fHx4dSpU1hbWwu2C0FBQc90w9bpdJSXl3P16lUmJyf5\nzne+Q3x8PHfv3mXnzp3C51VcXCyI883NzfHxxx/z2muvcfDgQQwGAykpKUxOTi7SfiotLeXBgwe8\n+eabqwrA5efnc+7cOfbt28fExATz8/O4uLjwgx/8gJ6eHhYWFgSF7JqaGgYGBjh06BCPHz/m0aNH\nZGZmCpYhgYGBbNq0iZqaGhITE7G3t8fW1pbbt2+zefPmZf+2mpqamJqaei49GoPBQGtrK9XV1Wzd\nuhU7Ozumpqbo7++ns7OTkZERqqurKS8vx0I6i+NEMiLD8+nurJVhtYKkMjnu7u4vFLgbbSa2bNmy\n5PxIpVKcnJwIDQ1l27ZteHp6IhaL6ezspK6ujvv37wuZG1tb2xUlAGxsbAgODiY7O5uBgYGvrZw8\nMjJCa2vrcwX73ybWm5bXeanExcXh5uZGa2vrorKL0TPIyckJMzMzmpubhcBBLBZTXl6OSqVicnJS\nMAS1t7fHwcFB+HnRxr+IiAi6urrIzs4WRosTExNxdHTkZz/7GVVVVdTX1yOXy4mKimL37t2IxWIS\nEhL44x//SFxcHGq1WvANc3Z2ZmZmhnfeeWdJAGNhYYGrqytqtZqcnBxycnIA2Lt370tb1Tk4OHDk\nyBGuXbvGe++9t6bzYhRkM34mMpmM8fFxwsPDsbW15fPPP2fbtm1s3rz5lZLIP3ToEP/+7/9OWFiY\nMHINT0qMGRkZ9Pb2cvbs2UUTcQ4ODoI/1Xvvvbfs8ej1eqqrq8nPz8fe3l7osTBuRyaTMT8/L4yw\n9/f3Cxk74zSYsTwZExPDxo0bycnJ4de//jX79+9Hr9dz79493nzzzVUtG7RaLZOTk1hZWZGcnMzk\n5CTvvfceP//5z5mbm1tSupydneX3v/899vb2eHp64ufnx7Fjx3B2dhaCIq1Wi5+fHzdv3mTr1q1o\ntVoUCgVZWVm4urqysLCAVqsV/h0eHhbsWZ5+3PjvSo8ZKSwspKSkBJlMhk6nE4YEdDodExMT1Dc2\nEyjTIPuaLitPb1/CTMP44IMPiI6OZufOnc8luGdqaoqzszNdXV2rSj+IxWK8vb3x9vbmtddeQ6VS\nUVdXR01NDbdu3SItLQ1HR0ciIyOJiIhY0rNjY2PD22+/TVJSEh999BFnz5596X09KpVqvXdnFdYD\nnnVeKiKRiPT0dBISEpicnGR+fh4TExOsra35h3/4Bx4/fgwgpIF1Oh07duwQbiZarZbR0VFUKpVQ\nmjCOzcpksiVBkPHm9qzVkrGefvDgwUWPR0ZGCm7VOp2OpqYmdu/eDTy5cQYEBFBUVISlpSWOjo58\n97vfZX5+nl/84heLbohPo9frcXZ2XtSv9P777/Pd73535abO5yQkJITu7m6Sk5O5cGFt6XwPDw/m\n5+cZHh5m586dggqvUqnknXfe4dNPP0WlUnH48OE/qR7P01hYWLBnzx5SU1N5++23EYvFtLS0kJqa\nysaNG/n+97+/7M0tLCyM8vJyGhoaFo1OGwwG6urqyMvLw9zcnMTERNzd3fmXf/kX3nnnHeF1CoVC\nEKScnp4WJpmqq6vp7+/nvffeW/T7FAoFr7/+Oj09PVy/fp2pqSnOnDnzzBLm2NgY1tbWXL9+HYPB\nwMWLF5HJZGRnZy/pgzMikUgYGhrCwsKCwsJC8vPzkUgkSKVSpFIpMpkMiUQi6M6Ym5sjkUiora0V\nxAyffq1CoRCyUzKZbNFzK/07MjLCH//4R3x9fTl79qywb729vYKXmLe3NwqFgs6OVqSuBni+trO1\nIZIisg4jLjyOoKAgbt++zW9+8xtee+215zLO3LBhA21tbWt+j9GENiEhgYSEBGZmZqivr6eyspKc\nnByys7OxsbEhLCyM2NhYIRNrYmLCuXPnyM7O5je/+Q2XLl16qROTIyMjr1ym9lViPeBZ56Xj5+dH\nZ2cnt2/fpqWlhYCAAA4dOoREIiEjI4OSkhJhFWhsEDZiTCF/+UvAYDAwPT0tBELGqRaVSsXMzAx2\ndnZLskL29vbPHIMPCwujsrKShYUFRCIR586dWxQ87dq1i9/97nfExMQIwY2JiQleXl60trYSGhq6\nZJs6nW7Zm3BlZSVKpfKlZVD279/PBx98wL1799i5c+eqr7W3t+e73/0u1dXVgmfW0xYJxtXnn2qU\ndjWio6Oprq7mwYMHqFQq2tvbOXHixLLWHUZ8fX1paGgQBAKlUinNzc3cvXsXiUTC4cOHBcPHyspK\n3NzcFgUnZmZmqNVqbG1thezO+Pg4mZmZXL58ecUMglqtZmFhQRAtjIiIYNeuXSteh/39/UJD/PHj\nx4VAczWnc4lEwhtvvEFsbKwQnCwX8NfV1VFQUMDFixcRiUT867/+q6BE/DQikYihoaE1T2np9XpS\nU1OJiYlZFNTDk7KOv78/GzZsoLi4GIVCwfd/8ENEd96Hyfo1bf+5kCrAYRPwpDR15swZmpubuXnz\nJl5eXhw4cGBNZS5fX19u3brF/v37X2g3zM3NiYuLIy4uDq1WS2trK6WlpRQVFXHv3j0sLCwIDAxk\ny5Yt2NnZMTw8zNTUFH/4wx84derUmixA1sLIyMiyhsPrPGG9h2edrwWxWExAQABbtmwhICBA+EL2\n8/OjqalJ8OIxrjjd3NxWDQSMMvq2tra4u7vj7+9PeHg48fHxxMfH4+3tjbm5OWq1mu7ubiorK7lz\n5w6PHj2ipaWFnp4exsbGmJ+fRyKRCB5KlpaWFBcXY29vT0REBPn5+QQHBwuaKqampoyPj/P48WPc\n3NyEDI1GoxEmX75MeXk59vb2dHV1AU++DM+ePUtDQwM1NTV4e3s/U+13refY19eX1NRUnJ2d19QQ\nbWpqSklJiWAN8PTNQCqVEhISwuDgIDk5Od/oKO1qGK+L27dvY2dnx8WLF5+Ztre0tCQvLw8PDw9a\nW1spLCykra2N3bt3c+DAAezs7ITtpqamsnXr1kUr4/r6elxdXbGzs6O+vh5TU1MePnwolK+Wo6Oj\ng6SkJC5cuEBkZKRgI5GVlSVkCJ++xmdmZrh27RoWFhZcunRpUdBia2tLUlLSsr1wbm5u/OIXv0Au\nlyORSFb8u3F0dKSxsRG1Wo2npycajYb29vYlgnRDQ0O0tbU9UyXaSGFhIdPT0+zdu5fCwkK2bt0q\nPGcwGOjp6aG8vJwjR46wffv2J8Ghuh9GSl5+47JIRLfzjxkemaS7u5v29naGh4c5ePAgKpWKW7du\noVAocHZ2XvX7xdLSktzc3GVtSZ4XsViMg4MD4eHh7NixA19fX2ZmZmhqahLKf8PDw8CTxVFLSwti\nsRgPD4+vvBgqKioiNDR0WY+/ddYDnnW+YUQiERs2bKC0tBSlUolOpxPUXp/2LXoeJBIJFhYWODk5\n4e3tTXBwMLGxsWzfvp3g4GDhRjYyMkJjYyP3798nLy+P+vp6YZpl48aNbNiwAZ1OJ/gjGVfxrq6u\n5Obm4uPjIwQ8VlZWyzY6arVasrKyhMyAr68v7777Lg4ODkRERDA7O0tKSgpmZma4uLh85S84ExMT\nXF1dSUpKIiQk5JmBlPHGLZFIMBgMS5yijZNgUqn0lfAFWlhYICsri0ePHuHv749eryciIuKZ7xOL\nxTQ1NTE+Pk53dzdbt24V+raePue9vb1UVlZy+PDhRY+3tLRgY2ODk5MTJSUlqNVqRCIRR44cWfYz\n6+7u5vPPP+fMmTPCNSKXywkKCsLNzY07d+7Q2NiIu7s7CoWCiYkJPvzwQ+RyOZs2bVqivG1ubk5z\nczNdXV1C0COXyzEzM+O//tf/yv379+nv7xeEAxUKxZIypEgkwsPDQzCgdXd3Jz09XcgMGRkfH6e+\nvv6ZWUKA4eFh0tLSuHDhAtbW1uTl5REfH49EIqG9vZ1PPvkEKysrLly4sDiTZBWIoelXiF5mwCOS\noPM4wy9vDFFTU0NLSwvt7e10dXWxefNmwsLC8PHx4e7du9TV1aFUKlcM4EUiEb29vYtsSV7KLv5H\n72JwcDDbtm0jPDyc8fFxhoaGhNcY9ZYyMjL4m7/5G1JTUzl48OALNWBnZ2eza9euV0o9/VViPeBZ\n5xvHzMyMqKgoNm/ejJmZGbW1tfj6+pKfn8/IyAienp4vxeVXJBJhamqKnZ2doEQbGRnJ1q1biYuL\nQ6lUYmpqikajoaenh7KyMlpaWlhYWODBgweMj48zNTWFSCSio6MDtVot3GzlcjlNTU3Y2NgImZUr\nV66we/dusrOz+eKLL/D19eUnP/mJcCMyGmT6+/uTm5tLc3Mz3t7eX1l92tbWFr1eT0FBAREREc/s\nZ+rr6xPsDOLj45e9gbu6ugq+QCYmJl+78etydHd388knn2BmZsaFCxcIDg4mPz8fKyurVTM8vb29\n3Lx5k6GhIezs7IiKiqKrq4uwsLAlrzU6XH/ZyLW9vR25/MnkT2ZmJuPj41y+fHnZz6q/v59PP/2U\nEydOLNsDYmNjQ0xMDHNzc9y4cYPR0VFyc3OJjY1lbGxMWJFrtVrq6upISUkhLy+PqKgo3n77bcbH\nx7G2tubcuXN8+OGHhIaGUlZWhkqloqOjg6qqKvr7+5c9PoVCgUajoba2lsjISAYHB5mdnV1U9piZ\nmaG6uvqZZpN6vZ6rV68SHx+Pr6+v4Avm5uZGQUEBxcXFHD16lK1bty4KqAwGA03tfaja8rGVqBCL\n9Kv+njUjMUW87QoevpHU19ej0+kwGAyYmZkRGhoqTF5GRUUxPz/PjRs30Gq1eHh4LPs3olar6erq\n+kpaWc/CzMyMhoaGRQEPPPkMfv7znzM0NCTICjyvSOvs7CwPHjx4KdOgf6msBzzr/EkwlpRcXFzw\n9/enqKgIDw8PpFIpGRkZa0pDfxWkUilWVla4uLiwYcMGQkJCiIuLE7JCY2NjdHZ2CvL+w8PDjI2N\n0djYSE9PD8PDw+j1egYGBggMDKSiooKTJ08yNTUlTMs0NDRgamq6KOUPT1bvkZGRjI2NkZqaiqWl\nJU5OTl/pWJVKpVC6e5am0czMDIODg0ilUuzt7VdscrS1tSUwMJCMjAxGR0eFnpevG61WS25uLnfu\n3OHAgQPC1I1YLMbJyYnU1NRFnlFGBgcHSUtL48GDB8TExLBjxw5KS0s5efIk9+7dw8bGZtGxzs7O\nkp6eTmJi4pIA2yggaWNjQ0FBASdPnlw26BsaGhJG1FfzLjJmW1xcXAR5gPDwcCorKwWRxd/97ne0\ntLQwMTGBWCzm0KFDbN26lYsXL/Lee+9x4MABrK2tsbGxob29ncnJSQwGA2KxmNOnT684gu/h4UFu\nbq5gEHr37t1F/loajYZHjx6RkJCw6udSXFzM+Pj4omxYdXU1paWlODg4cO7cuSW9d8PDwyQlJdHY\n2IjeMQFffRYSvoJTuhGJAkJ+Cp4nsbOzw8bGhtbWVsRiMRs2bODBgweCTYeDgwPe3t6EhoZSXl7O\n/fv3cXZ2xsbGhuTkZIaGhvD29sbMzIzc3FzBif7rYm5ujoWFBUJDQ9mxYwfHjh0jOjqaX/ziF+h0\nOvR6PW5ubiuaE6/EwMAAfX19z6Wn9G1jPeBZ50+OMQBoaWmhr6+PAwcO8ODBA6qrq19YW+NFEYlE\nKBQKQkJCmJycZGBggIsXL9La2irc8P39/ZmcnGRkZISWlhbu37/PL3/5Szo7Oxdta2FhgcbGRv7L\nf/kvLCws0N7ezv3798nMzMTf35/g4GC8vb3Jysri8ePHeHt7v3Aq2liKys7OFoLFlZBIJFRVVWFu\nbs7ExMSqtggKhYKwsDBKS0upq6sjMDBwRa2Rl0F/fz9XrlwB4OLFi0scp21sbBgaGqKrq0sI7FQq\nFenp6RQUFBAeHk5iYiJKpRJzc3OKi4vZuHEjHh4eZGRkEBMTI6zuS0pKMDMzWzYzMjAwwPT0NGVl\nZSwsLPD6668vec3o6Cgffvgh+/fvX7Z5/ct0dXXxxRdf8PrrrxMREUFmZiYzMzNERUVhbW1NbW0t\ns7NPzDXFYjFHjx5d8Vzb29tTV1eHSCRCKpUKjfTL3aiNIo4ZGRns3r2bqqoq7OzsFvV8FRcXrxrw\nqFQqbt68yYULFzAzM2Nubo5bt27R2dnJxo0bef311xft69zcHDk5OWRlZREVFcXx48cpq6zD1v8w\nNhNZ8FU0eSSmYBcHm38PoiefpbOzM2KxmJ6eHt59913i4+Oxtramrq6O27dvMzY2hoODA/Hx8Vha\nWpKamkpbWxutra10dXUREhKCnZ0djx49+tqFOF1dXYmIiGDDhg3Y2dkhkUiETFxhYSG2trZ88skn\nz+3g3t7ejkajeS47lm8b6wHPOq8EEolE8D3Kyspi9+7d2NnZkZKSwszMDEql8hsdlRaJRILzdmFh\nIbOzsxw5coTCwkJ27txJWFgYkZGR1NTUcP78ee7evUtz81KvIKlUiqmpKTk5OTQ0NNDX18f8/Dzb\nt2/HxMRESLn39/eTnp4uTJm9CFKpFC8vL5KSklY13lQoFBQUFKBWqxkbGyMmJmbVEqJMJiM0NJSe\nnh7u3r2Ln5/fS5/g0ul03Lt3T7gp79mzZ8Xgz8vLi/T0dGxtbSksLOTOnTsEBQVx6tQpwc8M/nP6\nSKvVEhYWRmdnp+B5ptfruXHjBvv371+2b2xkZISmpiZGR0cJCAhYkr2ZmJjggw8+YOfOnURFRT3z\n+FpaWrh+/TonT54kMDAQW1tbnJyc6OjooLT0iTWCubk5IyMjwBMRy6en6L6MtbU1TU1N+Pj4cP78\neYqLi6mrq8PPz2/Zz9Le3p7Ozk7Gxsbw8fGhpqZGCNKkUikFBQVs37592VKPXq/ns88+Y9OmTfj5\n+dHc3MyVK1dwcHAgODgYtVotnB+9Xk9FRQWff/45tra2nD17VlgoZGRksOvIm8hdt0F38n80MD/n\nrLpEAfabYHcGSBaXFz09Pdm0aRMymQyRSIS9vT2hoaFEREQwPj7OnTt3qK6uxtHRkV27dpGbm4tW\nq0Wv19Pf309UVBQqlYq5ublFuk7fFHv27OGnP/0pP/nJT5472IEnU3mWlpZLyrPr/CfrAc86rwwi\nkQg3Nzc2bNjArVu3MDU1JTExkebmZkHXwsHB4RurTxuzJgMDA3R2drJt2zZMTU2prq4WsiITExOM\njY0REBBAenr6oqkamUzGuXPniImJQaVSLXJt7+7upq+vj4mJCQwGg+DOfOvWLfr7+/H29n6hTIqF\nhQXm5uZkZGQQERGx7DZEIhGPHz9Gr9fj4OCAwWBYkkn5MmKxWMiopKSk4OHhsaqg3vMwPDzMlStX\nUKvVXLx4EU9Pz1U/Y+MkXlFREX5+fpw+fRpfX99lA+KFhQWampoIDQ3F3d2dlJQUQkNDhfO/Ulaj\np6eHiooKfHx88PLyWlTOmpqa4oMPPmDz5s1s2rTpmcdXW1vLrVu3OH/+PN7e3sLjHR0diEQiTp8+\nTV5eHh0dHRw9ehSNRiM0Ga9GREQEQUFBmJiYEB4eztDQELdv30apVC4bxCmVSlJSUti5cycFBQWE\nhoZiamqKSCQiPz+f2NjYZXuUHj58yMjICLt37+bWrVuUlpZy/PhxNm/ejEajoampicjISKFxe2ho\niBMnThAbGysErSMjIzQ0NDxpjLb0Bc/TMFQA2mnQryHbI5I+yeyE/gNs+u2SYMfIcteAMfu1efNm\nbG1tqa+vJzMzc5F44uTkpOAP9oc//IGxsTGCgoJeSi/h8yAWi1/4++3Ro0d4e3u/VF2fvzTWA551\nXjksLS2JiIigurqaiooKXnvtNXx8fMjKyqK1tRUPD49vTCPmaf+qvr4+9u3bJ0xsWVpaIpfLKS4u\n5vLly9TW1tLe3o5UKsXMzAylUsnVq1eJiYnBy8uLpqYmdDodzs7OHDhwAIPBwMDAAFVVVeTk5NDW\n1oajoyNjY2Pk5+djbm7+Qr09Li4uDA4OUldXR3Bw8LLvHxsbY3p6GoVCwePHj9dU9xeJRLi7u+Pk\n5MQXX3yBhYXFC61Ejej1eoqLi0lNTWXbtm0cPHhw1SmzmZkZ7t69S1paGv7+/kgkEpycnFYVi7Ow\nsCArK4utW7eiUChYWFigtraWrq4uIiMjl91/nU7HjRs3kMvlaLVaoqOjhQBidnaWDz/8kPDwcLZt\n2/bMYywtLSU3N5dLly4t6QGqqanBysqK6elpOjo6OHz4MAUFBVhYWLBp06ZnXuNP3xyNGUmjiKFx\neu/pz97U1BSDwUBlZSXe3t709vYyODhIaWkpQ0NDQiDp7e2NSCRCJBIxOjpKcnIymzdv5vr16zg7\nO3Pu3DkhCykSiSgqKqK/v5+ioiJ27drFwYMHlwRczc3N6HS6/5RxMLEH3++BxQaYaoWFSZCYYdAv\nIOyx2ASk5iCSwIa3YOvH4HFMKGM9LyKRCDs7O0JCQoTpNqNMhUQiobW1lb/7u7+jvLycrKwskpKS\nePvtt7/WEu7LpKCggNjY2GfaqXybWQ941nklMWrCaDQaUlJS8PPz48CBA0xNTZGSkoJOp8Pd3f0b\nMcibm5vj0aNHeHl5UVZWRmRkJBUVFYSHh2NlZUV+fj4bN27k8uXLHD58mODgYKHZNDIyEnjSexIV\nFUVvby/e3t5ER0fj4eFBYGAgMTExbNu2jYCAACwtLTExMWF+fp7S0lLu3btHU1MTvb29jI+Po9Vq\nkcvlz1x5+vr6UlxcLEylfBmDwSB4mqnVanx9fdfct2BnZ0dAQABpaWlMT0/j4+Pz3EHZ6OgoV69e\nZXR0lDfeeGPVhmi1Wk1BQQEpKSk4Ojpy+vRpgoOD8fLy4saNGwQHB68YHMjlcmpra3FxccHa2hp3\nd3fBzPVpob+nyc3NZW5uDq1Wy9jYGAcPHkQikTA3N8dHH32En58fCQkJzzzm+/fv8/DhQ958881l\nhQRLSkowMTHhwYMHfOc738Hb25uYmBgmJydf+Bp3dHQkKCiI7Oxsent7l2S+3N3dKSgoYGZmhsbG\nRtrb24WJocHBQYaHh4mLi+P//J//g0ajoaioCBMTEzo6Ojhx4gRxcXHC9rRaLaWlpTQ3N+Pn58fZ\ns2dX1NMyyk4suhZFYrAJA/8fgPcbYBeLyMIbtdyLun5zbCL+CmnYTyDmX0B5Akxe3Hj3y9y8eZPJ\nyUkkEgleXl4oFAquXbtGa2urYA48OztLXFwcfn5+L+33fl3o9Xqys7PZt2/fK6OS/iqyHvCs88oi\nEolQKpV4eHiQkpLC7OwsO3fuJDQ0lIqKCgoKCnBwcPjadWKmp6epr6/n8uXLPH78mL6+PlQqFe7u\n7tjY2DAyMoJarUapVOLq6kpcXBxtbW3U1z9RljU2k8rlciIjI5fNSBibpR0dHfHx8SE6OprNmzcz\nPj7O8PAwbm5uTE9PU1tbKwgqGm9WMzMzwgi+8eZonFa5ceMGSqVySfnJ3NycO3fuIBKJCAgIYGRk\nZIkmz2qYm5sTFhZGUVERzc3NBAQErOmL1mAwUFpaSnJyMrGxsbz22msrBivz8/MUFhaSlJSEtbU1\nJ0+eJCwsTCi7GN9XUlJCWFjYigHI2NgYExMTeHt7I5FIePz4MZOTk4KT+NN0dHSQn5/PqVOnKCkp\nwcrKivj4eDQaDZ988glubm4cOHBg1WDHYDAIPVtvvfXWiiJwOTk59Pb2cvHiRaHJXCwW4+XlRUhI\nCJWVlYLX11pEJY0oFAoiIiJoamri3r17bNiwQdCfGR4epri4mMnJySXvE4vFbN++HWtrax49eiSc\nJ29vby5duiTsgzFYvnr1KgaDAZ1Ot2IvlBHjGP6Kr5Hbgk0IuO5H5n2K+okNVPaYEbLpMIi/ellJ\nr9cLQwgdHR10dnYyNzeHTqdjdHSUiYkJRCIRVVVVwnukUimJiYlf65j6y2JsbIyGhoY1ZR2/zawH\nPOu88lhbWxMeHk5paSkVFRWEhIQQExODtbU1t27dore3F09Pz69NbGt8fJy2tjbi4uIICAigo6OD\nubk5uru7iYyMFOwJjNkceKK2PDExQW9vL9PT0/j5+T13FkQqlRIcHIytrS3379/H1dWVxMREdu7c\nSXBwMNbW1szNzfH48WNKSkq4c+cO9fX1dHd3MzY2hkQiQalUkpaWRnh4+KLzI5FIaGxsxN7eXvCI\nWkmTZyVkMhlhYWF0dHRw//59AgICVtUUmpiY4Nq1a/T29nLhwgUCAgKW/X1GHaQvvvgCU1NTTp48\nSWRk5LLlLnd3dx48eCAYQK5EZWUlUVFRLCwskJmZiaOjI/Pz84uaU2dnZ/nkk084fvw4bm5u3L17\nlw0bNhAQEMCnn36Kra0tR48eXfUc6fV60tLS6Ovr4/LlyyuWF/r6+iguLubs2bOL+nqMmJqaEhoa\niq2tLRkZGTx+/BgPD481K3QbhwAAkpOThWb4jz76iKmpqUWvNTc3F6xVzpw5Q3d3Nw0NDYJx6fDw\nMGFhYZibmzM8PExycjINDQ0cOXKEnTt30tbWhq2t7YrN9hqNhjt37nDo0KE1Z6s8PDzIy8vD1tZ2\nzd5QGo0GlUpFT08PLS2mihnZAAAgAElEQVQt1NTUUFJSQkFBgWAcPDAwwPz8PBqNhunpaWFC6uLF\ni+zatYv3338fkUiETCbD2dkZd3d3VCoVCoUCa2vrV1bfpqenh9HR0TWJcn6bWQ941vmzQC6XExoa\nyuTkJKmpqYJ+T3R0NIODg6SmpiKTyZb0LbwMhoeH6evrE4KbwMBAQRzMqOOTlZVFdHS0UGq6d+8e\nc3NzwgSIRqN5rgzK0xhVmmtqarh37x4eHh44Ojri4OAgZAM2bdrEli1bBAPI0dFRGhoaqKqqYmFh\ngdLSUqanp5menhbE2YwTQSMjI8hkMqysrJ7beFAsFhMYGMj8/Dypqal4enouWcUbDAaqqqq4du0a\noaGhHD9+fNnymbFEcu3aNUQiEYmJicTGxq7ayyIWi3F1dSUlJYWoqKhlS32WlpZkZ2cTFxdHfX09\nGo2GI0eOCF5Xcrkcg8FAcnIySqWSuLg4xGIxBQUFBAUFCQFVYmLiqjdsnU5HUlIS09PTXLx4ccXg\nZGxsTFBZPnbs2IrbgyfTVdHR0YyOjnLz5k2hwXwtgYOx58rLy0uYdjx27Bizs7MMDw8LAY27uzvj\n4+M4OTlhZWVFamoqWq0WsViMXC7nwIEDuLu7k5ubS2ZmpjBmbrxWent7AVb0cOru7mZoaOi59GEk\nEgmOjo6kpaURHR0tKINPT08LQwQNDQ1UVFRQVFREbm4uBQUFtLe3MzExATwpvfr5+REbG8u+ffvY\nvn07UVFRBAUFCTYbERERXLp0CXt7e2xtbbl06RJeXl4kJiby7//+72zZsoWZmRny8/MFD0B7e/tv\nvJn5WTQ3NwuSGeusjMhgvOrXWefPhPb2dqEksnPnTkQiEcPDw9y6dQuNRsNrr732zAmX56G+vp7a\n2tpFrtB6vZ7/9//+HyqVih//+MckJSUJSs4A//RP/4ROp0On0+Hn58euXbu+sqmfwWCgtraW27dv\nExcXx44dO55ZRjIYDExOTnL16lVMTEywsrJicHCQ0dFRFAoFer0etVpNeHg4MzMznD9//oUDxsbG\nRlJTUzly5IgwxTY9PU1qaioTExOcOHFi2SyMTqejqqqKgoICnJyc2L17N66urs/1uzMyMtBoNBw/\nfnzZ5z/66CNiY2O5d+8eu3fvxt/fn+zsbGZmZkhMTKSsrIzS0lLeeecdoUn1H//xH3F2dsbCwoKz\nZ8+ueq41Gg2ff/45MpmMU6dOrdjoOj09zR//+Ee8vb2ZmprijTfeWPMxjo6OCpoyR44cWdU89cvM\nzMyQlJSEXq/n1KlTjI2NceXKFebm5vD09OT+/fsA+Pj44ObmRnd3NzExMezZs4fGxkbu3r2Lv78/\ne/fuXRKsPnjwgNHRUY4cObLs775//z4zMzMcPHjwmfu5sLDA+Pg4Y2NjjI2N8ejRI/R6PVKplLGx\nMcFPb7kfS0vLNV+709PTjI2NrXn83GAw0N3dTVlZGU1NTcJiy9jg/acmLS0NJyenNU0NfptZD3jW\n+bNkamqKL774AplMxokTJzA3N8dgMFBdXU1OTg5BQUHs2bPnpUxzlZeX093dveRmqtVq+fnPf46D\ngwMxMTE0NTVx/vx5AJKSklAqlZSXl7N3796X2vg4NTXFzZs3hZv1WsZQZ2Zm+N3vfsfhw4cJCgpC\nq9XS1dXF1atXUSgUyOVyhoeHMTExwcXFBScnJ5ydnXF2dsbJyWnN5cKBgQGuXr1KVFQUdnZ2ZGZm\nEh0dza5du5YEDHq9ntraWvLz87G2tmb37t0vrH8yPz/P//2//5cTJ04sWyIqLCykt7eXgYEBfvSj\nHwnGtb/+9a/Zt28ft2/f5u233xbKMnq9nv/1v/4XLi4ui4Kg5VCr1Xz66afY29tz7NixFbMv8/Pz\nvP/++wQGBiKTyZienl5TEPA0BoOBpqYmYfz8wIEDWFparum9er2evLw8qqqqOHXqFA4ODrz11luk\npKQIU1kGg4GdO3fyr//6r1haWnL79m3EYjGHDx9e0V6kubmZkpISLl26tOzzV69eJSwsjJCQEAwG\nAzMzM0JA8+Wf2dlZbGxssLW1xcbGBgsLC4qLizl27Bj+/v6vhEeUWq2murqa8vJyYYovMjLyGxVI\n/TIffPAB27dvf+Es8reF9ZLWOn+WmJiYEBERwfDwMBkZGYIujIuLC1FRUXR0dJCRkYG5uflXtqho\nb29f1mhTLBZjZ2dHeXk5CwsLPH78mM2bNyORSNi4cSPu7u5Cg+dyruoviomJCWFhYYjFYpKTkwGe\n6bQsl8tRKpVcv36djRs3Ym5ujq2tLRUVFYSFhSGXywUhudDQUMHQsLKykuzsbKqqqnj8+DHDw8Oo\n1WokEomg4fI0FhYWgo5Sc3Mz586dIyoqalEQYDAYaGho4IsvvmBwcJCDBw+SkJDwlXR9pFIptra2\n3L59m+jo6CVBh1wuJzc3l61btwrmnkb5gLS0NPbv3y8EpQaDgRs3bjA0NMTRo0dXDSinp6f56KOP\nUCqVvPbaaysGO1qtlk8//RQXFxf2799PRUUFLi4uz+1RJhKJhAB7cHCQmzdvIhaLcXNze2aZSyQS\n4ePjg729PdevX+d//s//yZ07d9BoNIIdik6no7e3l4yMDExNTdm1axeHDh1atSHZYDBQUlIiCCUa\np9t6e3tpa2ujtLSU2dlZioqKyM7OpqysjN7eXtRqNXK5HBcXF2Gycd++fWzevJnw8HACAgLw9vbG\n3Nychw8fEhsb+0pkU2QyGR4eHsTGxuLm5kZbW5ugn2Vqaoqtre03vp+5ubmCTtg6K/PnITCwzjrL\nIBaL2bdvH0qlks8++4xt27YRHx+PmZkZr732GpGRkdy6dUvQ8lluNHgtzM3NrfhFsnHjRgoLC5mc\nnEQqldLa2rpI2j00NJQ7d+6suo0XQSQSER0djY+PDykpKTQ1NZGYmLjqNI+Hhwe7du3i888/5513\n3kEmkwlGrS0tLRw/flxQ3H16kkyv1zMyMsLg4CCDg4NUVlYyODiIWq3GyclpUTZocnKSrKwswsLC\nmJ6eJjs7m/Pnz2NhYYHBYKC1tZXc3FwAIch4WTeHoKAgqqqqhLLV01hYWKDRaJZMyA0MDCCTyYR+\nFuOEVU9PDxYWFouEJL/M+Pg4H330EREREezYsWPF49Dr9SQlJaFQKAQfqpGRka/UYCqTydi7dy+R\nkZGkp6dTWVnJkSNH1qSy6+/vj5ubG0VFRYvE94zMz8/T3d1NcHAw4eHhS543GAyCSvfY2BgjIyOM\nj4/z/vvvMz4+zvT0NFZWVtja2qJQKBCLxcTExAilp+f9O4iMjKSqqorS0tJXqmRjnCJVKpUcPHiQ\nmpoasrOzmZ+fJzo6mqioqG9EE2d+fh61Wv3ShED/klnP8KzzZ48xM5GXl0drayt+fn6COahxMicl\nJQW1Wo2Hh8dz61Q0NDRgY2OzbF+QSCTC1taWhoYGZDIZra2tbNq0Sbj5yWQy+vv70Wq1X4vjuJmZ\nGREREWg0GpKTk5HL5StqoQC4ubnR1dVFW1sbgYGBzM7OCk3VwcHBlJeXL7GlEIlEggjihg0bCAsL\nIz4+ntjYWKEnp6+vj3v37lFdXY1MJhMyRgaDgdzcXGQyGbdv36a5uZmEhAQOHjyIvb39S18Je3p6\nkpqaSkBAwKJjKCkpYWxsDDs7O6E/qK2tjcLCQk6fPk1GRgaRkZEUFxdTU1ODRCJBrVYLE3DOzs6L\n5A+Gh4f58MMP2bJlC1u3bl3xOAwGA+np6UxNTS3qA8rOzmbXrl1fuUSjUCgIDw9HoVCQlpbGwMAA\nSqVyxe1OTk6SnJzMlStXaGtrQ69f3rlcq9UK5Zr29nbq6uooKyvj/v375OTk8PDhQ3p6epienkYm\nk6FSqdi2bRu7du1i//79xMfHExERIfSxJSQkYGlp+UIifsbm65SUlEWyBK8SUqkUd3d3YmJiUCqV\ntLe3k56eTm9vL6amptjY2HxtWZ+hoSG6u7uJi4v7Wrb/l8R6wLPOXwSmpqZERETQ09NDdnY2np6e\nQhOju7s7ERER1NfXk5OTs+oI7XLU1NQIGYzlsLW1pbGxkaCgIJqbmxkdHSUwMHBR0FNSUkJ0dPRL\nOdYvY3TiDgwMJD8/n/r6ery9vZddST+tHC2VSoWG1Y0bNzI2NoaTkxMDAwNr6gWQyWTY2NgwPz/P\nw4cPCQoK4jvf+Q6hoaFYWloK00AjIyOC0q6XlxcymQytVouJiclLn3YxbrOwsFCYqtPr9SQnJxMR\nEUFfXx/BwcHMzMzwySefcOLECXx8fJiYmKC4uJje3l527txJRUUFOp0OjUbD/Pw8CQkJwvns7e3l\n448/Zv/+/c/8TPPy8nj8+DEXL14UgpDZ2VkePHjA7t27X8pNUCQS4eTkRExMDP39/aSlpS2ZWDQq\nLF+7do3AwEBUKhVFRUXP3K6HhwdarRZLS0s8PT2JjIwkISGBhIQEYmNjCQkJwdfXl7a2Nnx9fXF3\nd190TOXl5Tg6OgplxBfF3NwcjUazyNblVUQkEmFlZUVgYCBxcXFotVqKioq4f/8+Go0GW1vblx6w\ndXZ2Mjs7+0qfl1eF9YBnnb8YjH5P5ubmJCUlLZLXl8vlbNy4EScnJzIzM2lra0OpVK4pvV5eXo63\nt/eqI9v29vbcuXMHKysr1Go1fX19QtBjY2PD3bt3CQwMFATgvg4UCgWRkZGCGvVK/UsSiQQfHx+S\nkpIIDg6moqKCuLg4ysrK2LNnD1lZWWvS5NFoNNy+fZuioiJef/11Nm/eLPTFaLVaampqUKlU7N+/\nn02bNtHQ0ICFhQV6vZ66ujpyc3N5+PAh7e3tDA4OMjMzAzzJWn0VBW1XV1fKy8uBJxmt5uZmBgYG\n2L9/P9nZ2cTHx5OUlMSGDRuEUWmVSkVlZSWJiYkEBwdz7949oczl5eVFfHw88ESY8LPPPuP1119/\n5g3m4cOHVFRU8Oabby5qnh8YGKC/v/+5xrTXgkQiwdfXF39/f4qLi3n48CHm5uYMDg5y/fp1mpub\ncXFxYWhoiJqaGiEIXQ4zMzN+9KMf8cMf/pDAwEC8vLxwcnLC0tJSMOd8mq6uLiHL8TR5eXlERUWt\nKL74PCiVSvLy8gRPvVcdqVSKm5ubYC3T2dlJeno6PT09yOXyl9br09DQgJmZ2XNN7X1bWQ941vmL\nw8nJicDAQLKzs+nu7l4kr29ra0t0dDQTExOkpKSsSdekpKSEjRs3rtq4aW1tTUdHB3K5HFdXV4aH\nh+nu7iYgIACxWMzU1BRDQ0Or+j69DEQiEV5eXvj6+gr+XN7e3ktKHAqFAhsbG9LS0nB2dsbR0ZH6\n+no2bdpEW1vbMzV5urq6+Pjjj7G0tOTChQtCY+/Q0BC3bt2isLCQqKgoEhMTcXd3x87OjrCwMEpL\nS1EoFJw5c4YdO3YQEhIiZIm6urp49OgROTk51NXV0d3dzejoKBqNRiiTreUGYczq3bhxg4iICHJz\nc4mKisLLy4vKykomJycZGhri5MmTiMVioe9n27ZtlJeXExAQQGlpqRDwHDlyBHt7exobG7lx4wZn\nzpx55tRdbW0teXl5vPnmm0uum/b2djQazaJerxdlfn4elUpFd3c3LS0tVFdXU1dXx8TEBKOjo9TW\n1gqBZkREBD4+PiwsLDA3NydYjyyHTCbjww8/XPPkkUqlYmJiYtF5WVhYIDs7m0OHDr0UuwOxWIyT\nk9MibZ4/FywtLQkICGDTpk3odDoePHhAQUEB8/PzL9TX9DRlZWUolcpVhTfXecJ6wLPOXyTGbEdb\nWxt5eXn4+PgIX95G+f7g4GChL8HJyWnFVWhRURGRkZHPzM44Ojpy//59JiYmeOutt3j48CF9fX34\n+/tjaWlJTk4Omzdv/kYmOCwsLIiMjGR4eJi0tDRsbGyWNG07OTkxPj5OX1+f0POk1Wpxdnamqalp\n2ckyrVZLTk4Od+/e5fDhw2zfvh2pVMro6CgZGRncvXuXkJAQTp48iaen56JA0ujqXVtbS2lpKYGB\ngVhaWmJvby98HssJKDY2NpKXl0dxcTFtbW309/czNTWFwWDA1NR02RufhYUFarWa8vJy+vr6eP31\n1xGLxfT09FBdXc3ly5cxNzenvr6erKwsLl++TFBQEGVlZUxPTwNPPs/x8XESExOprq4mMzOTN954\n45mj821tbaSlpXHp0qVlMxG1tbVYWlquqcHYYDAwNTUliO3V19cLYnt37tzh3r17dHZ2MjExgVgs\nxt7eHj8/P4KCgpiYmMDU1BRvb2+6u7sZHh6murqagYEBxGIxb7/9NpmZmQBCpkcqlSKTyTh//jxb\nt25dc2ZmZmaGjo4OwsLChMf6+vro6+t7qY3Gtra2DAwM0NPT82fhcfVlJBIJrq6uwsBBV1cX6enp\ndHV1IZPJsLOze+7vh3v37hEZGbnqgmydJ6wHPOv8xWKU15fJZCQlJWFpabloFWRmZib0m6zW8Jmf\nn8+WLVue2WBqYWFBX18fw8PDBAcHExcXR1FREYODg0RGRlJdXS0oun4TGP20PD09ycjIoLu7Gx8f\nn0V9Mz4+PlRUVAg3pvr6enbt2kVGRgYxMTGLXtvX18fHH3+MVCrljTfewNXVlYmJCbKyssjKysLf\n359Tp07h4+Oz4upbIpEQEhLCyMgImZmZbNiwYUkWQSKRYGlpiaurK35+fkRERLB161bCw8Oxs7Nj\nYWGB3t5eysvLycnJoaamhq6uLlQqFXNzc0ilUkxMTPD09OT27dtCo/XCwgI5OTkoFAp27dpFS0uL\nEJgYS38uLi5kZGRgY2NDR0cHFRUV3L59m+7ubv7qr/7qme7wvb29fP7555w7d25F8ctHjx7h4+Mj\nBKALCwuMjIzQ09NDa2srNTU1PHr0SLBEqK6upr+/n7m5ORQKBW5uboSEhLBjxw727t3Lpk2bCAsL\nw8/PD3d3d6F0EhYWRmJiIm5ubjx8+JD5+XlhH1xdXXnrrbe4dOkSIpGIwcFBrK2tOXXqFB9++CEJ\nCQkkJSVhMBhQKpXPvAnr9XrKysoWBTf19fXI5fKXrv7r6enJrVu38PHxWbMG0auIhYUF/v7+wjl7\n+PAh+fn5qNXqNWd9DAYDWVlZ7N2798/G1f1Pybrw4DrfCgYHB7l27RpeXl4cPnx4yZeDRqMhPz+f\nyspKEhISiImJQSwWYzAY+Md//Ed++tOfrimFrlKp+O1vf8uWLVvYs2eP4LCtVCqxsbGhv7+fEydO\nfF2HuSILCwuC15ZRxM3I6Ogo//Zv/8aJEye4desWP/7xj0lJScHb25vY2Fh0Oh0FBQWUlZVx6NAh\nQkNDmZqa4t69e9TU1BAbG8vWrVufW+TRqPFz4sSJF16t63Q6VCqVMDI/NDTE4OAgGo0GR0dH+vr6\nkMvlnD17lrq6OmZmZmhvb+f06dPcuHGDCxcuLFHAvnjxIteuXRMc0iUSCTKZjP/+3/87q60PVSoV\nH3zwAUePHiUwMFB43Ci2Nzo6ytjYGFlZWXh4eAij3Wq1WhDbM/7Y2dkJ4ntrneQaGxvj5s2bLCws\ncPz4cUGXqLy8HLlcvijg2bhxI8eOHVv1MzN6n1laWnL8+PFVb8AajYZ//ud/5qc//akQHBkbpJcb\nbf+qVFZWUlJSwrvvvvuV+r1eNYaGhigrK6OmpgZ3d3eio6NXNeadnJzkd7/7HT/+8Y+/4T3982Q9\n4FnnW4PR72lkZIQzZ84sq1lj7EHRarX8/+ydd1SVd57/X7fApVx671Wp0gSxi72XGGMsiY6ps9My\nZTOzZ3fP2fOb2dn9YyY7ObNzJpPsZJIYo7GgFFEElCJFikgHCb33JnCB235/5NznhACKippyX+f4\nh9zn3qdceL6f51Pe7507d2Jra8sf//hH/vVf/3Xe+zl16hTd3d386le/Ar5UZv30009xcXGhvLyc\nX/ziF89stLaxsZGEhAS8vLzYunWrcBzvvfceIyMj2NraCpoymZmZ7N69m7i4OMzNzdm1axcSiYSc\nnByKi4sJCwtj9erVj6Uw29LSwvnz51mzZs2Clj7Gx8fJzs6mrq4OlUqFQqFgYmICS0tLFAoFKpWK\nNWvWsGTJkmnNo8XFxSxfvnxW/R0TExNOnjzJ888/P+O1gYEBPvroI8Hwc2BgYJpFgoGBgRDAVFVV\nsXPnTiHbZ25u/lhlTq1WS2FhIRkZGaxatYoVK1ZQUlLCtWvXUKvVglfW4sWLuX37Nj4+PlhZWVFV\nVSVo+cy1f5VKxbVr12hoaODgwYP37RN55513eO211wQ9mD/96U8cP378oZzeH+acT548iZ+fn9BQ\n/l1CqVRSVVVFcXExAwMDhIWFERERMSM73NDQQFZWFj/4wQ+ezYF+y9CXtPR8b5BKpQQEBAhjytbW\n1jP6WkxNTQkLC0MikRAfH8/AwAAjIyOsXLly3vtxcXEhOzsbf39/5HI5BgYGwuSPTqH4Yb2iFgor\nKyvCw8Opr68nLS1N0JfRGS5WVVVx8eJFbGxsaGtro7S0lLVr17J69WoKCgqIi4vD2tqaAwcOEBQU\n9Ng6MhYWFgQEBJCSkkJPTw8+Pj4L8sQulUq5fv06mzdvZunSpeTn57Nz504CAgKorKzE0tIStVpN\nbm4u6enp3L17l/b2dn73u9/R0NAw62cqlUrKy8tZu3Ytd+/epaSkhFu3bnHjxg3hu5VIJKjVaiwt\nLfH09CQ8PJwNGzawdu1aIiIicHBwoLa2lueeew5LS8tZ1aofhoGBAc6dO0dXVxeHDx/GwsKCjz/+\nmLKyMqFB3srKiiNHjhAQEMDU1BQxMTGEhITg5eUlZDWdnZ1nFckTi8UsXrwYIyOjWcvCX+Xu3bvC\n79PIyAiFhYVs3LjxifSs6Ubm4+LiCA4O/s4pDEskEkE1ftGiRYL6dUNDA1KpFBsbG8RiMXV1dWg0\nmmkZRT1zow949Hyv0E3weHp6kpCQwNDQEF5eXtMWWV0vR1hYGNXV1XR1dWFtbY29vf28bt4mJiaU\nl5fT1tYmmInqgp6ioiKamprmNfb9pJBKpfj5+WFjY0N8fDzDw8O4urpSXl7Of/3Xf1FeXk58fDyO\njo6sXbsWiURCbGwsGo2GJUuWsHnz5gXNUBkbGxMSEkJxcTFlZWX4+fk9dj9CW1sblZWVbN26lbi4\nOGxtbamvr6eyspI1a9bQ29vLiRMnWL58OVFRUTg6OiISifjDH/6AQqGY83P7+/uJiYnB0NAQBwcH\nfH19aW9vJzg4mGPHjhEaGsrixYtxc3PDzs4OuVw+7VxaW1sZGhp67DKPRqMhPz+fS5cuERERwaZN\nm7h8+TJpaWmC4Obdu3eJiYlhw4YNwii5j4+PUMYyMzMjPDwcrVZLQkICw8PDuLm5zXrtHR0d8fX1\n5cqVK/T19eHt7T0jMG1paRGmFOvr61EoFNOamBcaExMTlEolpaWlBAcHP7H9PGtMTU3x9fUVZB9u\n377N9evXaW5u5u2332Z4eJh9+/Y968P8VqAPePR8LzE3Nyc0NJSSkhIKCwvx8fGZ8ZRoYGAgNCK3\ntrZSXV2Ni4vLvLR0pFIpRUVFhIeHC8GBLujJyspCoVAseDPnw2JjY0NYWBhVVVWUlpZSUlJCbW2t\n4K3k5OSEVCqlubkZlUrF6OgonZ2drFq1asGPRSqVEhwcTFdXF2lpafj6+j6W8ev169fx8/MTDEO3\nb99OdnY2fn5+bN68mfT0dEJDQ5HJZEilUiwsLLC1teXdd99lfHz8vsepc5YfGxujtLQUmUzG0qVL\nkUgkyGSy+waytbW1SCSSx5ow6u/v5+zZs/T19XHo0CE6Ojo4e/Ys4+PjbN++neHhYXp7ezl69Che\nXl73PR6RSISzs7Mw0ZicnDynfpNutL28vJyCgoIZfzM9PT2Mjo7i7e3NnTt3sLa2ntck2uPg6upK\nVlYW5ubm3wptnsdBLBbj4OBAWFgYfn5+fPjhhyQnJ1NSUsJvfvObBRfx/C6iD3j0fG8xMDAgODgY\nhUJBfHw8dnZ2M7Rnuru7GRwc5Pjx40xMTBAXF8fExARubm73bWK2tbUlNzeX8fFx/P39hZ/LZDJG\nRkaorKwUlIefJbogTC6X09jYSHFxseCcvX//fqRSqaBHo9FoBINWU1PTBW8W1alA60xRnZ2dH0mw\nbnR0lOTkZFasWMHVq1fZs2cP586dIzo6mvLycpYsWUJ/f79QNoAve35OnTrFvXv3aG5unlWQTywW\n88ILL/CHP/wBLy8v7t69i0qlwsnJibt37wrlsYqKCurr6+no6GBoaIjJyUnEYjEymYySkhIcHBwe\nyWZEo9GQl5dHfHw8UVFRLFq0iLNnz9LQ0MCqVauIjIwkJSWFxYsXs2/fvofqrTIwMMDPzw93d3cy\nMjKoqKjA2dl5xmdIpVKCgoKYmJggPj4eBwcHoUfn3r17tLS0EBQURGZmJiEhIU98IlGnzZOYmEhE\nRMT3YlJJq9XS29vL4OAgw8PDHDp0iB07djzrw/pWoA949HyvEYlEuLu7C0J1CoUCDw8P4elW5xcU\nGBiIq6uroCNz48YNrK2t5xTnk0gkNDc3U1dXR3Bw8LRshbm5OdXV1fT29qJUKh9bdn8hsLW1paWl\nBW9vb1xcXNi2bRuvvPIK9vb2dHR0CEGQnZ0dd+/eJTU1lfr6egYGBtBoNJiami7YYuPs7IyTkxOx\nsbHT+p3Gx8fn9RR769YtTE1NuX37NitXruT69etER0ezevVq1Go1xcXF+Pj40NzcTEBAACMjI5w8\neRJfX1/MzMwoKChgcnKSr89zyOVyzpw5g52dHbm5uYyMjPDKK68QGBhIWFgYK1asYPny5Xh6emJu\nbo5SqaS7u5vq6mpycnJIT0+nt7eX8fFx+vr6GBoaYmpqCrFY/EBRxd7eXs6ePcvQ0BC7du0iJyeH\nW7du4eHhwfHjx6mvr6egoIDnn39+hjv9w6Arh6lUKuLj4xkdHZ1R5tL9zTg7O3Pp0iVUKhXu7u6o\n1WpKS0sJDw/n2ump/6kAACAASURBVLVrbN269akEIJaWloKf1LdRm2e+jI+PU1RURGxsLPn5+UxN\nTfHxxx+zZcuWZ31o3xr0AY8ePXx501yyZAn5+fmUlpbi6+uLoaEhTU1NqFQqofwkk8kIDAzEzs6O\n5ORkmpqa5rSoUKlU9Pf3MzAwMC3LI5fLKS0tJSYmhry8PNRq9QPF7J4UWq2WyspKzp8/j1KpxNnZ\nmf379wuaMLt27aKgoABPT0/6+vo4cuQIa9euZdmyZVhaWjI0NERZWRkpKSnU1NTQ19eHSqXCxMTk\nsVLsVlZW+Pn5ceXKFYaGhjAyMuK9997DycnpvgrQuoZ0U1NTTE1NqaurIyQkRCjDubq6kp2djbOz\nM6Wlpfj5+XHy5EnCw8NZu3YtN27cYOvWrYIasy7D4eLiQkpKCkFBQeTm5lJdXc3LL788o5dJIpEg\nl8uxs7PDw8MDf39/wsPDWblyJdHR0RQUFLBixQq0Wi3d3d1UVVWRnZ1NZmYmlZWVNDQ00NnZKQRD\nIpGIwsJCEhMTiYqKQqlUkpKSglQq5dixY3h7e/P5559jZGTEkSNHFqSso+tzCwsLo7a2lmvXrmFm\nZoadnd20oEz3N5OdnU1xcTFTU1PU1NRQWVnJxMTEDMf6J4lOm8fT0/Nbrc0zF0qlknfeeYeGhgah\nxywwMFDvn/WQ6MfS9ej5ChqNhszMTO7cucP+/ftpbGxEJBIRExMzY1uVSkVOTg75+fmsWrWK5cuX\nTytzjY6O8pe//AWxWMyJEyemTYTl5eXR3d3N+vXr+eSTT1i2bNlTHa/VarXU1taSnp6OVCpl/fr1\nWFlZ8dFHH/HLX/6S4eFh3nvvPUxNTbG2tsbf318wgZwtE6FSqWhvb6elpYWWlhZaW1sxNzfH3d0d\nDw8P3N3dhXHlh0GhUHDmzBna29vRaDS4urry6quvzrl9TU0NaWlpqFQqjIyM8PX1nTEp1NLSwoUL\nF4AvdXw2bNhAcHAw165d486dOwC8/fbbDAwM0NTURENDA62trRw4cIDx8XEyMzN55ZVXHlrZVqFQ\n8O677/Iv//IvM67h5OQkAwMDQoA8MDAg6ApptVpMTEwYHx9HJBIREhJCaGgoTU1NFBQUsG3bNpYs\nWfLEmuB1asAmJibs2LFjRlClVqv54x//yMTEhPAzU1PTp64NU1payq1bt3j99de/U9o8OkpLS0lI\nSECj0WBoaDgvPzc909FnePTo+QoikUhQwb148SITExM4OTnNmoERi8V4enoSEBBAYWEhubm50ywq\nDA0NuXv3Lh4eHtTV1U27OVlZWXH16lXWrFlDYGAgSUlJiMXiOdV5FwqtVktDQwOxsbE0NDSwfv16\nNm/ejLW1NUZGRuTl5REQEICVlRUODg5UV1czODhIZ2cnW7dunXNRFYvFWFpa4uHhQUhICCtXrsTN\nzQ2lUkl9fT03btygsLCQzs5OxsfHMTQ0xNjY+IGLtFar5datW0Ij8djYGAEBAXP2pyQmJjIwMIBc\nLsfDw4MtW7bM2IeFhQXNzc10dnYK5/rpp5/S0dEBfPm9bdiwQQjYysrK6O/vp6amhubmZo4fP/5I\nvSnd3d20t7cTGRk54zWpVIqZmRn29va4urrS19dHQ0MDUVFRDA0NMTo6ipeXFyEhIYyMjHDz5k0a\nGxuBL5uYm5qa6OzsZGRkhKmpKcEiYiGCIAsLCyIiIoS+HYVCgaurqxDc6xS9S0tLhTJgYGDgtKzm\n08DBwYGqqioUCsUzy5g+STo7O2lpaUGr1aJWq9m5c6e+Ufkh0Qc8evTMgrW1NcHBwWRnZzM4OEhg\nYOCcNxdjY2OWLFmCXC7n8uXLdHd3CxYVOsG7xsbGaVL4hoaGtLa2AuDp6Ymfnx+XL18WHJafBM3N\nzVy6dImqqirWrl3L9u3bsbW1FRZFkUhEe3s7UqkUR0dHrK2tqampISIigrt371JbW4u3t/e8pqdE\nIhFmZma4uroSFBTEypUrhbJgS0sLN2/eJCcnh7a2Nu7du4dEIsHU1HTGAt3a2kpBQQEGBgZoNBo0\nGg2Dg4P4+fnxj3/8g6NHj/LrX/+aP/3pT9TV1TE8PIyNjQ3u7u7s2rVr1gW/vr6eW7duCX1JOuNX\njUYDfOkx9lUn8+zsbMbHx9FoNNjY2LB27dpHuv4NDQ1MTk7e1zS0u7ub06dPMzo6ipGRERUVFdjZ\n2fHaa6+xbNkyJiYmKCwsZNmyZbz88stERUXh5uaGqakpExMTdHZ2UlFRQVZWFtnZ2VRVVdHY2EhX\nVxcjIyMolcpHCoZ0ujchISFUV1eTlpYmTLbpvmuVSiX8Tpubm+Pv7/9UDT6/y9o8tbW1JCcnc+LE\nCUJCQjAyMtJr7zwC+pKWHj334fTp04jFYrq6ujhw4MAMG4KvMzk5SUZGBmVlZaxfvx4PDw8+/fRT\nVq1aRV1dHUePHhW2raqqorCwkOPHjwNfish98sknrFu3joiIiAU7h/b2dtLT0+nv72fdunWEhITM\nmfK/desWfX197Nq1C4CmpiYSEhLw8/Ojq6uLnp4eNmzYQERExGNnD4aHh2lpaaG5uZmWlhZGRkZw\nc3PD3d1daCSXSqWo1Wo6OztpamqivLycgYEB4uPjqaysnDZCrhP++3//7//x61//etZzrKqq4sqV\nKxw8eJCBgQESEhL4zW9+Q1JSEuXl5QCEhYWxd+9e4MsM0+9///tpU1ubNm16pNH8tLQ0DAwMWLdu\n3YzX1Go1N2/epLCwECcnJxobGzExMWH//v14eXkxNTUlKB7v379/XhkMhUIxrUT21ZKZVqvFxsYG\na2tr4Z/u//ORXbhx4wZlZWXY2tqyfft2bGxsUKvVvPPOOygUCgIDAxkYGODgwYNPzTtOR1ZWFu3t\n7Rw6dOiZaV0tJK2trXz++eez2qDoeTi++zN8evQ8BpOTk2zYsEHoJdE17M51I5XJZGzdupWwsDAu\nX75MSUkJYrEYJycn8vLyaGlpEaayFi9eTFJSEkNDQ1haWmJtbc2xY8f45JNPEIlEhIeHP9axd3d3\nk56eTmdnJ2vWrCE8PPyBT9zu7u4UFxcL//f09MTKygqZTEZ/fz8vv/wyCQkJ1NTUsHv37sdyaLaw\nsGDJkiWCON34+LgQAKWkpNDb24uTk5PQAxQVFcXq1av5yU9+Qnl5+bSeEfgyaNAtur/61a9mBDzF\nxcVkZGTw0ksv4ejoiJubG8nJyVy4cIHOzk5kMpmQbdPR2tqKWq0WfNXs7OyEMfCHpb+/f1aBvM7O\nTuLj4xGLxajVapqamli/fj2rVq1CJBLR2dlJbGwsLi4uvPnmm/POXBgbG+Pq6jpjkdRqtSgUimkB\nUF1dHfn5+QwMDCASiaYFQF8NiHTZvYqKCu7du4eFhQV///vfWbZsGatWrUKtVpOfn09YWBhhYWF8\n+OGH7Nmzh8WLFz/09XpUVq1axd/+9jdqamrum037NtDX18fZs2fZt2+fPthZAPQBjx4990GhUGBk\nZISHhwcODg6cO3eOlpYW9uzZc1+1YQcHB1555RXBbfvatWusXLmSGzducPz4cUQikaBpUlpaKjz1\n29jYcOzYMU6ePIlYLCY0NPShj7mvr4+MjAyamppYvXo1Bw4cmPd4sKOjI8PDwygUCmFxW79+PefP\nn0culzM6Osqrr77KzZs3ef/999m6deuCNcyamJjg7+8v9H5MTk7S1tZGc3Mz2dnZdHR0YG5uzt//\n/vdpRphfZ3JykoSEhGmeV7m5uUI2TTflJRKJCAoKori4mI0bN5Kfn8+hQ4eE94yNjREXF8eiRYtY\nuXIlp0+f5ujRo3z00Uc0NDQI1g3zpb+/f1rjts6UtaCgAENDQ0ZGRggICOC5557DwMAArVZLTk4O\nubm5QmPyQiASiTAxMcHExGTWYGh8fHxaMFRbWyv8XyKRYGVlxdDQEFqtlvb2doyNjWltbWXPnj1k\nZGQwOTlJZmYmv/3tbzl48CAXLlwgLCyMmJiYp9JMLJFI2LVrF7GxsXh5eX1rS1sjIyOcOnWKTZs2\nPXOR0u8K+h4ePXruw82bN4mOjkYmk2FsbExYWBjNzc1cv34dDw+PWf2HdIhEIpycnHB0dKS4uJj2\n9nbGx8ext7cXxNpMTU1JT0+fljUyMTHB19eXS5cu3de76OvonLivX79OQEAA+/fvx8PD46EWGZFI\nRGNjI+bm5kJgYG5uTlNTEyYmJvT09BAUFISnpydeXl6kpKTQ1NSEp6fnY/tqfR2pVIq1tTVeXl6C\nzk1vby+XLl2a1dxTx9TUFLa2tuzcuROtVsuNGzeoqKiY0Wys1WrJy8tjfHyckZERbGxsBMuHyclJ\nPv30UwICAtizZw+Wlpa0t7cLAn0pKSksXbp03tdWo9GQkpLC7du3EYvFGBgYcOrUKdra2hgfH8fc\n3JwTJ04QFRWFRCJhZGSEc+fO0dfXx9GjR5+aQKVIJMLQ0FAQmPTy8iIwMJClS5eyevVqwsLCBONR\nrVaLVqtlamqKsbExPv/8c6HEqFQqqaio4Le//S1Llizh1q1bVFRUsGjRoqfSaGtpaUlvby/Nzc3f\nymBhYmJCkEtYSFPd7zvfvdk9PXoWEF2GR4dUKmXXrl3ExMTw6aefUlxcPEOg7uv4+voikUjYsmUL\nYrGYCxcu0NfXByDYN7S0tEx7j52dHS+//DIpKSlUVFTc9/NHRka4fPky//d//4e5uTk//elPWb16\n9SMHIG5ubjOOZ/369bS0tPDFF18IpSRnZ2feeOMNrKyshBLCk0TX0D2fRlidQvSVK1eor6/nxIkT\nM8pvpaWlwqj32NiYELyoVCrOnj2Ls7PzNDkCPz8/7t69i5+fH5aWluTl5ZGQkEBMTAxOTk7Y2tri\n6+vLv//7v9Pe3j5tX4ODg0JTdHp6Ov/4xz8YGBhgamqKF154gR/96EfCuHd1dTUffPCBICr4KGrT\nTwKRSIRcLsfU1BSNRoOBgQFBQUEcOXKE3/zmNzMm53QPA3K5nJdffhknJyc++OAD2transrxbt68\nmerq6qe2v4VCpVLx+eef4+np+URsXL7P6DM8evTMgUql4ubNm6xfv35GycbBwYHFixeTkpJCR0cH\n3t7ecy7EIpFIKAfs37+fwsJCCgsLUalUuLq6olaraWxsnDF1YWpqire3NxcvXsTS0nKGs/vY2Bjp\n6ekkJSXh6urKgQMHWLx48WOr22o0GsrLy6f1EOk8xdRqNYaGhsIkmVgsxsfHB1dXV5KSkujs7MTT\n03NBFHYVCgUlJSX09PQI5ZWpqSk++eQTpqam5nyfXC7nn//5n6msrGRwcJCXXnppxmTZwMAAFy5c\n4PDhwwwODqJSqejp6SE0NFSYltu7d++0DI6ZmRnXrl1jxYoVVFdXc/DgQS5dusQXX3zB6OgoCoWC\nwcFBCgoK+POf/0xZWZkwOpydnS1MMMGX2aXo6GiOHz+Ovb098GVmKikpieLiYl544QVCQkK+kU23\nJiYmeHt7s2PHDoKDgwXnbgcHB5KSkgTD1NOnT+Pl5QV8+Tfg7e2NpaUlsbGxwu/Qkzw/AwMDzMzM\nuH79+mOpTz9NNBoNsbGxyGQy9uzZ8438/r/N6AMePXrmYHx8nJKSkjmfskxNTQkNDaW2tpasrCy8\nvLzmnHARi8XcuXOHpUuXYmNjQ1tbGyKRiIyMDPz9/cnLyyM6OnpG0CSXy/Hy8uLixYtYW1tja2uL\nQqEgKyuL+Ph47O3tef755wkICFiwUoGpqSkpKSmsWLFi2iJhb29PXl4ew8PD08a24csG5PDwcJqa\nmrh27dq0st2jMjQ0xOnTp2loaKCmpoby8nK++OILlixZQklJCSqVasZ7RCIR9vb2ODs709/fj6Gh\nIdXV1dTU1GBjY4OZmRlqtZozZ86wbNky/Pz8GB8fp7q6msDAQDIzM5FIJBw8eHDGd6HTVcrMzOTn\nP/85CoVi1sBL1zzd0NBAXFwcBw8e5OLFizO227p1q9DT097ezqlTpzA3N+fQoUOPfe2eJBKJBEtL\nyxkBhFKpJCIigjfffJPf/e53s/Yc2dra4u/vT1paGm1tbfj4+DzR0XV7e3tqamoYHx//xmvzaLVa\nrl69yr1793jhhRee6kj/94VvfsirR88zYmJi4oGaM4aGhuzbt4/o6Gg++ugjYbT563h5edHd3c3Y\n2BiLFy/GyMiIgIAAtm/fTnp6OhKJZNp01FdxcnLiyJEjJCYmEhsby//+7/8yPj7Om2++yfbt2xdc\nSl8mk2FjY0NnZ+e0n9va2uLn50dfX59QkvsqhoaG7Ny5kz179pCQkEBSUtJ9MzEPwtzcHAsLC5RK\nJSqVColEwpEjR/jv//5v1q5dO6Np3NjYGGtra/7pn/4JCwsL1Go13d3dQilO97ScmZmJsbGx0Bvh\n7OyMUqnExMSEoaEhoqOj58xQqdVq/u3f/k2Q978fExMT1NbWcvDgwVlfr6+vR6PRcPPmTU6fPs36\n9evZt2/ffZvhv8m0tbURFhbGxo0bBf+z2bCxsRHUsj/88EP6+/uf2DGJRCJ27txJdnY2Q0NDT2w/\nC8HNmzdpbW3lxRdf/F6YoD4L9BkePXrmQGct8CBNHJFIhLOzMz4+Ply5coW+vj68vb2ZmJggJSVF\n6OHp7OxEq9Xi5OSEpaUlqampbNmyhcjISDo7O4VpHWdn52lPz0qlksrKSpqbm+nu7mbr1q3ExMQ8\n0emT3t5eJiYmZhibOjo6UlBQADCnUaO1tTXh4eHU1tZy/fp1HB0dH6oP5d69e9y8eZO4uDhsbGwY\nGxtDIpGwbt06YbRep1rd19cnXLMf//jHbNy4kSVLlnDo0CHKysqEaS4zMzMiIyPp6enhxo0bHD16\nVAgs+vr6KC8vZ2RkhO3bt3P9+nUiIiJmfcJ+++23qaurm/e5qFQqurq6WLVqFbt27WLfvn1s2rSJ\nmJgYrKysOHv2rFB2+yaYyD4O169fZ/ny5fdt5NchkUjw9/dHJBJx6dIlrK2tZ5RsFwpjY2M0Gg23\nb98mODj4G1kmKi4uJj8/n+PHj89LB0nPo6EPePTomYPu7m66u7uFyZ0HIZfLCQ0NpaKigtzcXCoq\nKqitrcXKygpHR0fUajU1NTUEBwdjZWVFTU0NWq0WV1dX/Pz8yM/Pp7u7mzt37uDo6IipqSlFRUWc\nP38esVjMvn37CA4OJjExUVBCflJMTk4KTu9fxdjYWDC91OnEzIZUKsXf3x8LCwvBdftBE2M9PT2k\npaVx9epV7Ozs2Lt3L8uXL6ekpAR3d3e2bduGSCRCpVJx/vx5Nm/ejLe3N2FhYbz//vuCUeimTZuQ\nSCSCLYRYLCY8PJykpCTKysrYsWPHtHHsjIwMent7iY6OJjo6mtbWVnp6emaMnbe3t/P2229PEyGc\nLxMTE7z11lvI5XJEIhGVlZV8/vnnhISEsHv37nmpV3+TUSgUZGZm3td+5OvoHhQ8PT1JSEjg3r17\neHl5PZGAxNXVlZycHExNTYWeqW8Kd+/eJSUlhWPHjn1jGtS/q+gDHj165qCjo4ORkRECAwPn/R6p\nVEpgYCDV1dXCpFN7ezvR0dFYWFiQnJws9OrY2Nhw9epVoqKikEql1NXV0d3djZWVFQUFBWRnZ6NU\nKtm7dy/R0dGYmppiYWGBm5sbFy5cwMnJ6Ymp2BoZGXH9+nVWrlw5YwFyc3MjNzcXGxubB47M29ra\nEhYWRnl5OVlZWbi6uk4rwWm1WhobG0lKSiI3NxdfX1+ee+45AgICBK+tkJCQaerQt27dQq1W09vb\ny/DwMJOTk5SXl7NixQrWrFkjHK+ZmZngdxYTE0NraytarZaqqiqkUqmgaJyeno6fnx/37t0jODgY\nd3d3EhMT8fX1nZatOHnypGBM+jBoNBr6+vr47W9/KzQml5SUcPDgwW9sxuFhaW5uZmho6JHEMs3N\nzQkJCaGoqIg7d+7g6+u74BIHuqbqxMREwsPDvzElo9bWVmJjYzl8+PC85Sf0PDr6Hh49eubg6yPp\n86WtrY2Ghgbh/2NjY5SXl2NsbIylpSV/+ctfOH36NDKZDAcHB27fvk1fX58wPtve3o61tTWLFi2i\nu7tbKIXpcHd35+DBg8TGxtLU1PTY5zkb5ubmGBoaztpfYW5ujoeHBxkZGfP6LBMTEw4cOMDatWv5\n7LPPSE9PZ2pqirKyMt5//32Sk5MJCgrirbfeYs2aNTOyHcbGxkKwo1AoyMnJITw8nM7OTqFkJZVK\nZ3WbX716NatXr6asrIzBwUFef/11jh07xhdffMGf//xnzp07h1wuJyoqiqamJtRqNXK5nPXr13P5\n8uVp1727u3tevTuzoRNRfP/99wF48803n7hR7NOkra3tsc7HxMSEI0eO4OHhwQcffDBDFmEhcHd3\nZ/HixVy/fn3BP/tR6O3t5ezZszz33HPfqd+FbzL6DI8ePXNQX18vOEE/DGq1Go1Gg1qtZnJyUpjY\nKSws5O233yYjI4OkpCT+53/+BwcHBwYHByktLZ2mHuzk5MSLL76Il5cXWVlZlJaW4uzsLGQcLC0t\ncXZ25sKFC7i6uj6RVHhHR4cgnvh1XF1dycrKIiAgYF49GyKRCAcHB/z8/MjIyCAtLY2RkRHWr1/P\n1q1bcXJymtfYcEZGBhYWFgwMDNDR0SFo22i1Wry9vWe1uvjqCLq5uTlyuRxnZ2eKioqQyWSMjIwQ\nGRlJa2srDg4OWFhY4OTkRElJCWq1WliMcnJy5h3kfR2JRIKTkxNbt25lzZo137kJnOzsbPz9/QUt\noUdBJBLh5eWFra0tFy9eRCwW4+LisqAZMDc3N65evYqbm9s01eunzcjICCdPnmTjxo0PlUHW83jo\nAx49eubg7t27mJiYPHQzqbGxMYsXLyYyMpI1a9YQGhpKQ0MD//Iv/8LExAQqlYqpqSlUKhUVFRXY\n2toil8uRSCSCg7darSY6OhozMzPCw8PRarXExcUxOjqKm5sbUqkUKysrnJycuHDhAu7u7gt+Ax8b\nG6OtrU2wevgqJiYmVFRU0NTUNK8yxvDwMFlZWVy9elV40m5oaMDa2hpXV9d5LWrDw8NcvnyZvXv3\nkpSUhFqtFt6n0WgwMjLCx8dn2nt0I+hRUVHCeYyMjPDJJ5+wceNGIiMjqa+vp7y8XDAq9fX1ZWxs\nDBMTE1JTUwXn7ZaWFlJTUx9p8szc3JxTp059J/2QtFotycnJbNq0aUFKUTY2NgQEBJCenk5TUxM+\nPj4LVoIyMDDA3NyctLQ0IiIinok2j0Kh4OTJk0RERBAVFfXU9/99Rh/w6NEzBxUVFdjY2Nx3xPZB\niEQijI2Neeutt6aVuXQolUpGRkZYuXIlv/jFL4iKiqKkpISf/vSnQhZA19wZFhZGbW0tKSkpWFhY\nYGtri7W1NQ4ODly4cAEPD4/HMvP8OhKJhNzcXKKjo2d93djYmMLCQoKDg+ecLOns7CQ1NZWUlBQc\nHR3Zt28f4eHheHl5ERQURF5eHnfu3MHDw+OBjbvJycl4eXnxxRdf0NPTw6uvvoq9vT2WlpYcO3YM\nHx+fGQtYZmYmCoWC7du3IxKJhMUmPDyc6Oho6urqkEgkHD16lI6ODkpKStBoNLS3t5OamopKpeLW\nrVvk5uaiVCpJSUl56KZlAwMD3njjDcGB/btGf38/1dXVrF27dsE+09jYmNDQUKHHysvLa4aS86Ni\nZ2dHbW0t9+7de+qTcUqlktOnT+Pu7j6roKmeJ4u+h0ePnjmYnJxcsNHv2traOV9rb29n8eLF5OXl\nYW5ujpOTE3fv3p2xnampKXv37mX//v1kZGRw+vRpBgYG8PX1Ze/evZw5c2aGpcHjYG9vz/j4OKOj\no7O+HhgYiFQqJSUlZdrPtVotdXV1nDx5kjNnzuDg4MBbb701TWgPEAKVoKAgPvzwQwoKCua06eju\n7qaurg6ZTEZ5eTk7duzAxcVF+I5kMtmMMlFzczPFxcXs27cPkUiEUqnkzJkz+Pj4CGKSnZ2dODo6\nIpPJ2Lt3LxKJhL6+Pm7fvi14RcGXC9XY2BgbNmx46GyDVqvltddee6j3fJtoa2t7IpkrqVTK7t27\nWblyJR9//PEDLVbmi0gkYseOHeTm5jI4OLggnzkfNBoNFy9exNzc/KGm2fQsHPqAR4+eOXjUpuXZ\nuN+TpKOjI+vWraOwsJCxsTHCwsIoLS2dc3sPDw/efPNNPD09+fvf/05mZibe3t7s2bOHM2fOzBAM\nfFREIhFubm7TLBG+ikQiISwsjMbGRnp6elCpVJSUlPC3v/2NtLQ0QkNDeeutt1i1atWc11EkEhEd\nHc2JEycoKyvj1KlTDA8Pz9guLS0NZ2dnbt26Jag6w9xB6cTEBJcuXWL37t3I5XLUajXnz5/HysqK\nLVu2CItNV1cXjo6OwrFYW1szMjKCUqmcFnxJpVJ+8pOfCH5l88XIyIhVq1Y9FcPMZ0Vra+sTLdWF\nh4fz0ksvcePGDZKTkx9JFuDrWFlZsXLlSpKSkh7ohbcQ6HzdJicn2bt3rz7YeUboS1p69MxBfn4+\nwcHBC6JkbGVlxdWrV2e4fJuYmPDSSy/R0tKCRCKhoqICmUxGVVUVYWFhc6ruisVi3N3dCQ4OpqSk\nhMzMTPz8/PDx8SE2NhZvb+95NRM/iKGhIfr6+uYUGTQ3N6ekpIT6+npyc3MZGxsjJiaGzZs34+jo\nOO8eCRMTE8LCwrh37x7x8fGYmpri4OCASCSioaGBvLw8tFot9vb2hISECAtsVVWV0MCtQ6vVkpCQ\ngIODAytWrBD+r1Qqef7554Vj0mg0XLt2TSitJSQkoNVqEYlEWFpaCg3nugzS4OAg0dHRbNu2jbNn\nzzI1NXXfxdLExITDhw+zYsUKvvjiC1QqFe7u7t+5xS49PZ2oqKgFLad+HTMzM0JCQigpKaGoqAgf\nH5/HVqR2cXEhLy8PY2PjJ67Nk5WVRWNjIy+99NKCj9zrmT/6gEePnjnIyclh6dKlCyIKFxgYyNDQ\nEMXFxUiluRNWzAAAIABJREFUUsRiMSKRiIMHD/LrX/+agoICJiYmGBsbo7W1FZlMhkwme2CPgZGR\nkSBkeOXKFSYmJggLCyMhIQEfH58FCXru3Lkzq9r00NAQRUVFtLa2Mjk5yY4dO9i8eTPW1taPtKiL\nRCI8PDzw9fUlLS2N+vp63Nzc+PTTTzExMeHgwYNkZmayb98+oaxUUVGBg4PDNA2TsrIyqqqqBD+s\n1NRUurq6OHz4sJBpGRwcJCMjg/b2diFD4ePjQ1dXF4ODg8TExLBjxw7c3NzYtWsXKpWKgoICuru7\nWb58OceOHaOhoYH6+noMDQ0FbR6do7izszN/+MMfOHz4MMPDwxw5coTs7Gyqq6vx9fX9zmR8Jicn\nuXHjBtu2bXviDcAGBgYEBwczNjZGYmLiY+tQicViHB0dSUhIIDw8/Il9J7dv36awsFCvovwNQB/w\n6NEzBxkZGaxevXpBboQikYgtW7bw0ksv4eHhwYYNG9i2bRt79uwhNDQU+LIXQpcxWLFiBZWVlURG\nRs4reLCxsWHp0qX09vYKAn4ZGRn4+vo+VrOnqakpqampLF++XOiR6ejo4Nq1a0KZycfHh8nJSQYG\nBuatSn0/5HI54eHh9PT0EBsbi1ar5ac//Sl37tzBysqKgIAAYds7d+7g7u6OjY0N8GUgc/78eQ4d\nOoSFhQU5OTlUV1dz7NgxtFotZWVlXLt2jezsbMRiMebm5oSHh3P79m1GRkaIiYmhoaGBDRs2YGZm\nhq2trSBeaGpqSmlpKfn5+VhYWPDLX/4SAwMDpFIpvr6+hIeHs3HjRn7/+9/zzjvvEBYWJnh4BQcH\nExISQk9PD8nJybi4uDzTseiForW1lb6+vhlmsk8KXVDs6OjIxYsX0Wg0uLm5PXLWzMLCgsHBQerr\n6/Hz81vgo4WamhpSU1M5fvz4d+L7/rbzzZCb1KPnG4ZGo2FqamrB/ao8PT356U9/CnzZP3L69GmW\nLVvGmjVrqKioYGBgAICioiLUajV1dXUsWrRoXp9tYGDAhg0bCAkJ4cqVK0ilUj7++GNeeeWVR9ZH\nMTAwwMHBgba2NpRKJbm5uYLB5u7du5HJZIyPj5OXl8fY2BgtLS0LMvmiVqsFHSATExMSEhJobm7m\n2LFj07b7ag+Pril09erVODo6UlJSQmFhIWvXriUxMZGGhga8vb1ZsWIFbm5unDt3jq6uLmQyGfv2\n7ROOu7Kykvr6+hmLeGRkJDU1NTg6OtLS0kJWVhZisZhXX32V/fv3z3oefX19gkeURCJh8+bNuLu7\nc/bsWVavXk10dPS3usT1uIKDj4q3tzevv/4658+fp729nb179z7y3+rGjRv561//umC/uzpaWlpI\nTEzk6NGjT9QGRs/80Wd49OiZhYmJCW7fvs3q1auf2D7kcjkNDQ2CuJ2Hhwc1NTWEh4cjl8uRSqXc\nunVLGEufr1idiYkJISEhyOVyamtrKSoqmlOU70GoVCqqq6spKCigq6uLZcuWsWvXLtzd3YWykoGB\nAR0dHdjY2FBdXU1oaOhjLeIKhYLPPvsMtVqNra0tr7zyCmVlZfT19eHj4yNkc+BLm4klS5Ygl8vJ\nzMxkfHyc7du3k5eXR0pKClqtltHRUQICAti9ezfe3t5UVlaSmJjI5OSkIHz41afvyclJ6uvrBYNS\nHbom7suXL6NQKHBzc0OlUtHW1oanp+esT/B5eXn4+/tPW/BsbW0JCAggIyOD+vr6BdWZedrk5uay\naNGiZ+JPZWRkRGhoKK2traSlpeHp6flIJVypVIq5uTmpqakLps3T09PD6dOn2b9/Px4eHo/9eXoW\nBv2Ulh49szAxMfFE3ch1rF69mtzcXDQaDQ4ODvzyl78UyioxMTEYGBjQ1tbGX//6VyorK+c9USIS\niViyZAk///nPcXV15cMPPyQnJ2fe7x8fHycrK4t3330XhUKBpaUlr7/+OkuWLJk18AoLC2NgYIDR\n0dFZ9Ybmy7179/j4449xcnJiaGiIzZs3Y2BggFKpZNWqVVy9epX4+HhBlXpiYgKZTEZLSwsFBQU4\nODjw7rvvkpaWRkhICG+88QYnTpzAw8OD1NRU3nvvPTQaDW+88QYajWZWUUVvb28aGxsFFWcdGo2G\n0tJSNBoNpqamHDp0iB//+McYGhpy9uxZzp8/P2PMub+/f9bsmpWVFSdOnEAul/PBBx8s2GTd00Sr\n1dLW1oabm9szOwaJRMKOHTtYt24dJ0+evO904/0IDAzE0tKS3Nzcxz6m4eFhPvvsM7Zs2TJDCFPP\ns0Wf4dGjZxYGBwdpaGggMjLyie7HwsKCiooKjIyMsLe3RyQSCU/7tbW1yGQyIiIiCA4O5saNG1RU\nVODo6DjvyTGpVEpoaCgqlYqbN29SW1uLm5vbnH09AwMDpKenk5iYiFwuZ9euXSxdupSMjIz7uqNb\nWlqSnp5OdHQ0+fn5REREPHSWZ3BwkE8++UTIEMnlciIjI+nu7iY/P58XX3yRiIgIGhoaSElJwcHB\ngdLSUmQyGQkJCYhEIkxNTenr6+PFF19kxYoVDAwMkJycTEZGBt7e3uzfv5+AgACmpqYoLi4mJiZm\nxnHKZDJKS0txcXERsmIDAwOcOXOG8fFxjh49Snl5OUZGRjg6OjI8PCwI4yUkJDA2NoaLiwtqtZqb\nN2+yadOmWa+FWCxm0aJFmJqacvHiRWQyGU5OTt+aEtfQ0BClpaXExMQ860PBwcEBX19frly5Qm9v\nL97e3g+VqdFl7+Lj4wXj2kdBJ2wZGRn5xO8deh4efcCjR88s9Pb20tnZKTQUPyl0i3RWVhZLly4V\nFjsnJydSUlIIDAykrq6ONWvWEBERgVarJTExkd7eXlxdXec94urt7Y2RkRHV1dWUlpYyPj6Om5ub\nkK1pa2sjOTmZ9PR03Nzc2LdvHyEhIZiammJoaEhJSQkeHh5zlgzEYjH37t1DrVbT19eHubn5Q/UN\n9fT08Omnn7Jq1SqCg4NJSEjgwIEDGBkZkZmZiYeHB97e3kKD8OTkJElJSSiVSuFavPjii2RlZbFt\n2zYMDAxITEzk9u3bBAcHs2/fPnx8fIQG9ObmZoaHh+f8fvv7+xkbG8Pd3Z2ysjIuXLhAZGQk27dv\nx9jYGCcnJ2G6RyaTUVxczHPPPUdISAh1dXVcvXqVyclJFArFA+0D7O3t8fPzIzU1ldbWVnx8fL4V\nXlt1dXVMTU3NKP09K+RyOWFhYVRUVHDr1i18fX0fKktrZGSESCQiPz+fkJCQhw48lUoln332GR4e\nHqxfv/5hD1/PU0Af8OjRMwudnZ0MDAw8lZu5jY0NBQUF2NjYCL0eEokEqVRKQ0MDbW1tgiaPs7Mz\nERERdHZ2kpiYiFarxdnZeV5Psy4uLkK/jYGBAWlpaYyNjZGRkUFJSYkQGCxatGiGxonOsf1+Dapm\nZmakpaWxadMmMjIypgVw96OtrU0oAYSHh5OSkoKrqytBQUFMTk6SkJDAnj176O3t5ebNmyQkJDA1\nNUVERARNTU2oVCo2btxIQkICnp6e1NTUUF9fL/QbfTWw01FRUYGJicmcxrAajYaSkhIaGhqoqanh\n8OHD+Pv7C+djbm7O0NAQdXV1REVFkZaWRlhYGGZmZvj5+eHt7U1OTg7Dw8PY29tjY2Nz32uh0yGq\nr68nIyNjQa0UnhS3b9/G3t7+mZa0vo5UKiUwMJDJyUni4+NxcHB4qIZhnbilTCabJnXwIDQaDRcu\nXMDU1JRdu3Z9a7J03zf0AY8ePbPQ0tKCQqF4IqOqX0ckEmFoaEhBQQFhYWHCzx0cHMjMzMTJyQml\nUiksLFKpFB8fHwICArhz5w7p6elCRuVBN1pXV1fUajUVFRWIRCKam5sxMjLi0KFDLFq0aM7MgkKh\noKmp6b7OznK5nPLychYtWkRLSwsymeyBzawNDQ2cO3eOvXv3EhgYSG9vL2lpabzwwgsYGBiQnZ3N\nyMgId+7coaamBjc3N7Zv386KFSsQiUSUlJSwZs0akpOTmZqaQiqVsm7dOrZu3Tqr8GF6ejp5eXm0\ntLRgZ2eHg4PDrFmywcFBsrOzcXd35+DBg7M2fLu7u5OSkoKLiwsTExNoNBrBd00ul3Pv3j1MTU0p\nKyujuroaBweH+5YiJRIJ/v7+GBgYcPHiRczMzB5q0X3aZGRkEBER8Y0bt9aVp5ydnYmLi2NqagoP\nD495BSFisXha9m4+khRarZakpCTGx8d54YUXnokhqZ75oQ949OiZhcbGRrRa7VNrOrSzsyMzMxNX\nV1dhcRWLxchkMurq6mhoaKC8vHxaX4CxsTHBwcHY29tz/fp1Kisr77uojo2NkZubS0FBASYmJqjV\nat544w0AEhIShGmx2W7YhoaGZGVlsWLFivueh0ql4osvvmD58uWkpaXdV0eourqa+Ph4Dh48KFzn\nxMRE/P39GR4eJjk5mfLyclxdXdm6dSubNm3C09MTY2NjNBoNZ86cYWpqiubmZuRyOXZ2dkxOTgqZ\nFvhy4kon8ghf6qKUl5czPj5OR0cHOTk5REVFCQubWq0mPT2dzMxMLC0tWbZsmTBW/nWkUimWlpZc\nu3aNsLAw7t69S3BwsPB6UVERfn5+7NixA5VKJZQinZ2d76sS7OTkhK+vL1evXqW7u3tWU9RnjVKp\nJDU1la1bt35jy2+WlpYEBwcLWkzzFXz8avZuPg88mZmZNDc3c/To0e+MoOR3FX3Ao0fPLNTW1mJk\nZPTURkrFYjFisZiysjJh0VSpVLS0tFBeXo5SqWR0dJR169bNCCCsrKxYunQparWaxMRE+vr6cHFx\nETIXfX193Lhxg6SkJCwsLNi9ezcxMTGoVCpSU1PZtm0b4eHhFBcXc/PmTWxtbWco2BobG5OdnU1w\ncPB9+yKsra25cuUKmzdvFkT3dF5VX6WkpISUlBSOHj0qZJ1ycnIoKSmhvb0dkUiEu7s7o6Oj/OAH\nP8DCwkI475GRET777DM6OzsFIbo33niD8PBwDA0NuXTpEmq1GicnJ/72t78xODgoaBlJJBJqampQ\nq9VIJBKWLVsmTGoNDAxw+vRppqamOHr0KGq1mp6enjltNeDLQLWhoQGtVktVVdU0gcabN28SHh6O\nubk5Li4uRERE0NXVRWJiIkql8r5SA3K5nNDQUKqqqsjJycHb23tBFL8Xivb2djo7Ox/Yn/Sskclk\nhISE0NXVxbVr1/Dw8JhXw7+7uzvXrl3D2dkZS0vLObcrKiqiqKiIY8eO6VWUvwXoAx49emZB59H0\nNEXV7O3tSUlJESZ3ioqKuHbt2rRt1q5dO2vGRKfVExERQXt7OwkJCQwPD3P79m2ysrLw9PTkueee\nIzg4WLgxe3h4oFAoSEtLIyIiQihPJCUl0d7ejru7uxA0iUQi2traMDQ0vG+ZRdcjpNFoCAoKIjk5\nmaioqGkZilu3bpGdnc3LL7+MUqkU+nKampoICgripZdeIjQ0lPz8fIKCgoTvoK+vj9TUVK5cucLI\nyAiurq5MTEzw5ptvIpFIhOBqyZIlFBQUcP36dcbHx+nq6iI4OBhjY2PMzMzIyspCq9Xi6OjIgQMH\ngC8DsNjYWKKjo9m6dSsymUzQQXrQou7u7s7ly5exsbERSotarZaUlBQ2bdokTN1JpVK8vb0JCgqi\nsrKS1NRUjIyMBM+wryOVSgkICECj0XDp0iWsrKzmzDY9bSorKzE0NJy3KOazRCwW4+vri5mZGbGx\nsULT+f34avZuLm2e6upq0tLS9CrK3yL0AY8ePbNQVlaGo6PjU+2hkEgkqFQqamtr8ff3F5qR29ra\nBE2YlStX3lekTiwWMzU1RXd3N01NTYyNjbFt2zaWLVs2a6+Kp6cno6OjXL9+naCgIJycnIiIiKC7\nu5vExEQMDAyEUemRkRF6enoeuMhJpVIKCwuJiYmhvr6eqakpXFxc0Gq1ZGRkUFxcTEBAAJmZmUJf\nzqJFixgcHOTIkSMYGBgwMjJCamoqe/fupauri6tXr5KVlYWHhwf9/f34+PgI01lft7OQyWRYWlpS\nVFSERqNBo9EIthdisZjS0lKmpqZ47bXX0Gq1xMXFUVtby+HDh1m8eLEQfMjlctLT0wkNDb1vCUoX\nHDU2NqJWq/Hz82N4eJiKigrWrFkzY3sjIyMCAwNxc3MjOzubwsJCrK2tZ/WFEolEuLi44OnpyeXL\nlxkaGsLLy+uZl7jy8vLw9vb+RvcYfR3dNFxycjKdnZ0PnIaztbWlvr6e4eHhGZne5uZmLl26xNGj\nR5+J6KKeR0Mf8OjRMwvFxcV4eXk9dUl4e3t7rly5wpIlS4SSmo+PD1VVVahUKhwcHGa9wU5NTXH7\n9m1iY2Pp7e1l9erV7Nu3D2dnZ9LT06mqqsLR0XHWsXJPT09GRkZIT08nMDAQIyMjvL29Wbx4Mbm5\nuRQWFuLo6IiFhQWFhYUP1BexsrIiPT0dPz8/3NzcuHLlCoGBgZw9e5aqqiq0Wi12dnbExMSwadMm\n3N3diYuLY8uWLYKKck5ODhKJhNLSUoqLiwkJCWHfvn2Ul5ej1WoZGBggKioKpVI5a5/FhQsXGB0d\nRSKRCAGPg4MDzc3NlJeXExERgZmZGadOncLFxYUXXnhhRqlDJBLR2dkpmEzeD2dnZ0pLS2lpaWH1\n6tV0dHTQ399/X1kDnY+XiYkJycnJNDY24uTkNGtpxNzcnNDQUEpKSigoKMDHx+epCGPOxbVr11i3\nbt03qsw2H3TTcDU1NeTk5ODj4zPnOejKqvHx8fj7+wvfS09PD2fOnGH//v0LakWh58mjD3j06JmF\ngoIC/P39H8mO4XEwMDBgfHyclpYWIZNibm5OZGQkmZmZvPvuu8TFxQnjz6Ojo2RnZ3Pp0iUAtmzZ\nQkxMDHZ2dohEIqG/R9c029/fP0O/RyQS4eXlxdDQEJmZmQQFBWFgYICpqSmhoaEYGBiQkJCAUqmk\nubmZqKioB2aZ7t27R09PDzKZjNraWm7evMnk5CQ7d+5k7969+Pn5YW5ujkgkmmbcqTP4TEtLQ6vV\nsmLFCmG0vKqqiqKiIiYnJzl+/Dh9fX2o1epZe2z8/f3x8fHBzc0NGxsbFAoFOTk5/PCHPyQ3N5fP\nP/8chULB8ePHWbZs2ZwZk4mJCRobG6cZls6GLhPzn//5n1RUVAjj7osXL37g++zt7YmMjGRsbIz4\n+HiGh4dxdnaekZHTuYVPTEwQHx+PnZ3dNJuNp8XIyAiFhYVs2LDhWzl+LZFICAgIQK1WExcXh62t\n7ZzX0cjICLFYzK1btwgJCWFkZISTJ0+yZcuWWVW69Xyz0Qc8evTMQk5OjvD0/bSxt7fn8uXLRERE\nCFMfVVVVvPbaazQ1NfHFF18QFxfH0NAQlZWVWFtbs2fPHiIjI2ftJdAtxhEREbS1tZGYmIhIJMLJ\nyUlY6EUiEd7e3vT393Pz5k2CgoKQSqVCX0x4eDhNTU10dHSgVqvx9vaedbHTarW0t7dTX19PcXEx\nQ0NDaDQaJicn+dnPfoabm9u04GJqaopz586xc+dOamtriY2Npa2tDblczo9+9CNhtHxoaIgzZ86g\n1Wp5+eWXsbOzo66uDqlUiqen54zjMDAwwNLSEicnJ3x8fFi2bBk/+9nP6OjoQKlUCj5dOiPXuTA2\nNubGjRvCGPz9kEql/PCHP6SkpISVK1fi4uIyb40asViMm5ubcJ2TkpIAZmgs6bIOLi4uxMXFMT4+\njqen51MNPOrr61EoFCxZsuSp7XOhEYlEuLq6CurKCoViztF1FxcX8vPzAbh69SpRUVFPzR1ez8Ki\nD3j06JmFrKwsVq5cOW8l44VEJpMxODhIb28vXl5eAPzHf/wHt27dErZRqVR0dnbyySefEBQUhLGx\nMQqFgt7eXkxNTedsgvX19cXf35/i4uL/396dB0V9Zgsf/3Y3OwjKJiIgi4IoAiKKiFFxTXDfEo0x\nxjWZOEtqbtX8cetWTd26VXeZWzNz506cyWRPNIlBMeK+gAuyiYK4scoiIJvsCDQ03f3+4aXfIKig\nLNqeT1WqkqaXx27M7/RzznMOCQkJ2NnZGZriKRQKfHx8qK6uJikpiUmTJnUbEOrr60tDQwPZ2dnc\nuHEDKysrQ3qtvr6etLQ0jh49Sm5uLj4+PjQ3N6PT6XBxccHJyYmmpqYetRDnz5+nubmZzMxMtFot\nS5YsobS0lIiICMNz63Q6vvnmG1pbW3n77bdxc3MDIDc3F2tr6z4HFfv376e0tNTwXkRERLBy5con\nPsbS0pKrV6/i5eX11MGUycnJeHt787vf/Y7s7Gyam5vx9PTsV9BsamrKhAkT8PPz49q1a1y4cIER\nI0YYduy6jBw5ksDAQNLS0rh+/To+Pj5PrDMaSBkZGTg4OBjFUEw7OzsCAwNJTU3l1q1bjB8/vsff\n+a5duIMHDzJp0iTmz58/TKsVz0sCHiEeodfrOXfuHJGRkcNWHOro6Mjx48cJDQ3FxMSE2NhYMjIy\nut1nzJgxjBo1ipSUFM6dO8elS5dIT08nJCTkifUdVlZWTJkyBUdHR+Li4sjJyTHU9ygUCsaPH09l\nZSUpKSmGnZ6Ojg5OnjxJfn4+LS0ttLa2kpubaxjamZyczMiRI5k3bx4LFizA2dmZtLQ0ALZs2cLo\n0aM5evSoYdeqqyj58uXLjBs3jlWrVhEWFoZarSYtLa1bt9pTp05RUFDA2rVru6Wvbt++jYODw1NP\n3HSZNWsW0dHRaDQaPDw82L17NxMmTHjqZ3z//n3UavUTA6t79+5x6tQpdu/ejb+/P1euXCEgIIAT\nJ07Q1taGm5tbv/rVdH1Go0eP5vz581y/fh1nZ+duKVYzMzOmTJlCY2Mjx48fx8XFpdfC54GWkJBA\nUFDQkLzWUDAzMyMwMJCamhpOnTrVrRcWPAy4T548CTwsZJdU1svrxepmJcQLQKPRoFKphrWhmoOD\nA15eXmRkZNDU1NStrw48vCD+6le/oqKigurqakOnX0tLy6fuRHTx8fHhgw8+wN/fn71793Ls2DFa\nWlpQKBRERUXh7Oxs6EsDD3dUHjx40O057t+/T319PcuWLePBgwdkZWXR3NzMV199hb+/Py0tLXR0\ndODg4MDEiROJi4sjNjaWv//974aRGevXrzcct05LS2PatGmGICQ3N9dQL/Lohaa9vb1fuxoTJ06k\noqKCiooKLl68yM2bN/nDH/7A+fPnaWlpeeL79KQJ8B0dHRw6dIioqChsbW3p6Oigra2NefPm8cEH\nH9Dc3MyePXv6Ne2+i5eXFzt37mTatGlER0dz8OBBGhoaDD9XKpXMmzePlStXcujQIcOR+8Gi1Wqp\nrKzE1dV10F5jOCiVShYuXMjrr7/ODz/8wJUrV9Dr9ej1eo4dO4ZWq2Xr1q0UFhZSXFw83MsVz0h2\neIR4xIMHD7h58+ZTuwoPtlGjRnHkyBGuXbvG7NmzeffddykrK2PMmDH8/ve/Z8eOHYSGhnLnzh3U\najV6vR5zc3OSkpKoqKigs7MTW1vbJ3Z/7arvmTp1KqWlpRw7dsxwm5+fHyUlJYZj062trdy/fx94\neIHYtWsXCxcuZMyYMZw6dYrS0lIqKiq4evUqYWFhzJ8/n3v37qHVauns7CQ3N5esrCx8fHxYsGAB\nV69eZcOGDYZATq1Wc+zYMVauXImZmRmNjY18+eWXBAQEsHjx4h5rT09P7/dJOoVCYejHk5SURGdn\nJ2VlZVy+fBmNRtPrbK0RI0Zw+vRpwsPDe90NOnnyJDY2NsyZMwd4eIqnq7jb3Nwcf39/XF1diYuL\nIzs7m7Fjx/YrzdVVRzVt2jRqa2s5cuQIbW1tuLq6GlKO9vb2hq7CWVlZfe4q3F8VFRWUlpYyc+bM\nAX/uF4GjoyP+/v7ExcVRVlbGvXv3KCsrY9OmTVhYWDBq1ChOnTr12N484sUmAY8Qj2hsbDQMhRwu\n7e3tXLx4kaqqKkJDQ4mMjGTcuHG8++67bNu2zXDc2cTEhMDAQAoLC2lqauLtt982nHbKzc3l9OnT\n5Ofn09raipWVFZaWlr3W95iamjJ+/Hj8/PwMzQq7LqY5OTncuHEDPz8/AgMDyc/Px8TEhDfeeAN4\nGJhptVqKiorQ6XRotVqUSiVTpkzh/v37JCQkcOfOHYKDg7G1tUWpVJKXl4ejoyNBQUHk5OTQ1NTE\nnTt3MDU1JTg4GI1Gw549exg5ciSbN2/udc3Pc5JOoVBQWFhIY2Mjer0epVJJQEBArzsXJiYm5OXl\n9dorJy8vj7S0NDZs2GB4v4qLi2ltbe02eHbkyJFMmzbNcMLq0Wn1faFSqfD09CQwMJC8vDxOnTrV\nrU9SV1fh6upqTp48iZub24A3xMvOzkalUg3JjLnhYmlpSVBQECkpKeTk5LBu3TpDUO3o6EhRURF1\ndXW9FsuLF5sEPEI8ora2ltLSUqZOnTosr19WVsa+ffsYNWoUc+fOJTU19YkzqVQqFYGBgVhbWzNp\n0iTMzc0ZM2YMAQEBhIWFYWtrS1lZGefOnSM9PZ2GhgZMTEwYMWJEj2+pCoUCnU5HeXk5169fp76+\nnsWLF6NSqaitrWXhwoU8ePAAjUbDpEmTDLszBw4coL293fA8dXV1ZGZm0tDQQHt7O++99x4TJkzA\nycmJw4cPc//+faqrq5k1axbHjh0jOTmZwsJCxo0bh6urK9988w0tLS18+OGHjz0Cn5KSQnBw8DOf\npGtubqa4uBilUsno0aOJiop67Hvc2NhIXV1dtx2glpYWvv/+e9asWYOjo6Ph9uzsbCwtLQ0F5z9/\nb93c3AxB3tmzZ7G1te1RkPw05ubmTJw4EW9vby5fvkxSUhIjR47E3t4epVKJj48P9vb2xMTEoFQq\ncXNzG7BTXF01V32tm3pZ5eXlcfv2bcLDwzl9+jT29vaGtKu7uztHjhzBz89Pxkm8ZCTgEeIRlZWV\n1NTUDPmxW51OR0JCAmfOnGHJkiXMnj0be3t7bt68iaWl5RPHCjzuwqZSqXBwcMDX15eZM2fi4eFB\nQ0Nku1KJAAAgAElEQVQDaWlpnDt3jqqqKjQaDdXV1Vy8eJFTp06hUCgIDw9n+fLlmJmZce7cOUaP\nHm1Y348//siePXv4z//8T2JiYnBzc+Pu3bs9gie9Xm+4ONbV1aFQKPjmm2/o7Ow0rDkiIoL8/Hxq\na2vR6/WUl5eTnp5Oc3MzW7ZseWK66tKlS4SFhT3z6SRzc3Pu3bvHli1bKCoqoqqq6rFzs1QqFVeu\nXDEcR9br9cTExODt7U1ISEi3+6anp+Pu7v7YLsRmZmY90lyurq5YW1v3a/02NjYEBgZiZ2fH2bNn\nycvLY/To0djY2ODo6MikSZO4cOECd+7cYfz48U/sndRXZ8+eZfbs2UZ9oS8uLubw4cNs2rSJgIAA\nPD09OXLkCM3NzXh5eWFpaYlKpSI5OZnJkydz5coV7OzshuyUnHh2EvAI8YiysjJaWlqG9DRGfX09\n+/fvp7m52TBQEx7uClhZWXHp0iVCQkKe65u6QqFgxIgReHp6EhISgpOTEwUFBVy5coWcnBx0Oh3T\np08nLCzMkG7p6t9TUlLC7du3+fd//3du3bqFVqtFp9NRVVVFbGws69evp6mpCYVCYbiwWlpa4u7u\nzo0bNygvL2f69OncuHGjW8ATHh5OaWkplZWVKBQKQ4pnzpw5Tw04z58/z5w5c565uNzGxoZp06Zh\nZWWFr68vZ8+exczMrNfdCxsbG+Lj4w0DSjMyMigqKmLt2rU9Ar1Lly4Z0ndP0pXmam9vN6S53Nzc\n+hWYKBQKHB0dmTZtGh0dHYbhsa6urtjZ2REUFERZWRlnz57Fw8OjT4MzH+fBgwekpKSwaNGil7Lh\nYF9UVVWxf/9+1q1bZziVZ2trS2BgIFevXuXatWuMHz8eT09PEhMTSUhIIC8vD1dXVxkx8RKQgEeI\nRxQXF9PZ2fnEKdkDRa/Xc/PmTQ4cOEBISAhLly7tcaTcwcGBy5cv9zrFvL9+3i+nsLCQiRMnsnz5\nchYsWIC9vT1lZWXEx8dz7do1GhsbMTU1xd7eHl9fX65fv86JEycMAUuXrpqdrVu3MmXKFHx8fPDx\n8WHChAnMmjULX19f0tPTKSgoQKlU4uTkxIMHD9DpdIaAp7y8HCsrK5RKJc7OzqxcufKJF1WtVsvF\nixeJjIwckIuvqakpPj4+HDp0CA8Pjx61L0qlktLSUszMzFCpVMTExPD222/3OBHX29DQJ/l5mis3\nN5czZ8702nfnaZRKpSE4LS8v5+jRo3R2duLu7s7EiROxtrbm0KFDhnTns7xnRUVFNDc3P3Fcxsus\noaGBvXv3smTJkh41Sl1drhsaGjh27BgFBQVUV1cbTnS6ubkZvqSIF9fz73EKYWTa2tqGZE6RWq3m\n+PHjVFZWsnnz5sfOa1IoFERERJCYmNjrKaKnaWtr4/bt29y4cYPa2lomT57MmjVrcHV17Xbh8/Pz\nw8/Pz5BaysvL48SJEzQ3NzN+/HgSExO71el00el0ZGZmMmPGjF5f39nZmfnz55OWlkZTUxMTJkzA\n39+fM2fOUFJSQlNTEzY2NoZjwOvWrXvqBbnrSPpA7jQ4OjqyatUqoqOj2bFjR4+gx9vbmzt37pCa\nmmoY3/GopqYmzM3N+53esLGxYdWqVZSUlHDixAkyMjJ44403+j0d3cLCgkWLFhEaGkp8fDwff/wx\n8+fPJzAwEBcXFw4cOEBJSQnLli3rd1PN0tJSo72ot7a28t133xEeHk5AQECv91EqlURGRuLq6kp0\ndLThdq1WS1NT01AtVTwHOVcnxCPUavWgBzx3797lk08+wdLSkl27dj11OOWUKVOora3l3r17fXp+\nrVZLTk4O0dHR/OUvf6GoqIiIiAh++9vfEhUVxdixYx8bLHQdS4+MjOSDDz5g165dhvqgx+1ajB07\n9onrmThxIvX19axduxatVktiYiJJSUmEhISwZcsWjh8/jkqlYtmyZX06ddXfHjx9NWHCBMLDw9m/\nf7+h/1AXHx8fsrOzsbCweOwJvtra2ueab+Xh4cGuXbvw8/Pj66+/5syZM70GmU8zatQo1q1bx7p1\n67h69SqfffYZDx48YMeOHahUKj777DNDi4G+unfvnlEGPB0dHfzwww/4+fn16bi9n58fv/rVr7C1\ntTWkU/v691IMLwl4hHiEWq0etCnQWq2W+Ph4Dh48SFRUFFFRUX3ql6JSqZg1axZJSUmPvY9er6e0\ntJTjx4/zxz/+kZSUFHx8fPjoo49Yv349fn5+z1TvYmdnx/Tp0/mf//mfXncFLCws2Lp1Kzqd7rHP\ncevWLWxsbGhvb2f58uW4u7uTkJCAWq2mo6ODxMREbGxsmDRpUp/W1N7ePmhBaXh4OKNHjyY2NrZb\nE7/W1lY0Gg2vvfbaY4PFmpqabie2noVSqSQsLIxf/OIXtLa2smfPHm7duvVMDQXd3d3Ztm0bs2bN\nIjY2lpiYGGbPns2sWbP4+uuvuX79ep+ep+vk3tMC25eNTqfj4MGDODg4sGDBgj4/buTIkezevZsp\nU6ZgZmZGZWUlR44cQa1WD+JqxfOSlJYQjxisHZ6amhoOHTqEjY0N77//fp87IneZOnUqCQkJPS6q\n9fX13Lhxgxs3bqBQKAgMDGTnzp0D3vrfxcWF2NhY1q1bZ7j4tre3s3HjRtrb2/nTn/7EhAkT8PX1\nxcfHxxActba2kpKSwty5c8nMzDQMZTUzMzPsXigUCkaOHElqaipqtZrGxkaampqYOnVqrykGtVo9\naKdiFAoFy5Yt4+uvv+bSpUvMmTOH9vZ2Dh8+zLhx46isrHzsHKmampoBm2D+aJorPT2dN954o9/F\nsQqFgoCAACZOnMjly5f54osvmDJlCm+++SZHjx6lpKSE119/HVNTU2pqarCwsOjxu1ldXY2tre2g\nfREYDnq9nqNHj6LX61m+fHm/06MmJiasXLkSe3t7Fi9ejEKhYOHChRw5cmSQViyelwQ8QjxioGt4\n9Ho9GRkZxMfHExkZ+cSeOk9iZmbGjBkzSEpKYvHixX2qyxloCxcupKqqivPnz9PW1sacOXMMF/j6\n+nry8vK4evUqhw8fxsPDA19fXyoqKvD392fatGkkJCRQV1fHwoULcXFxMXRhnjZtGuXl5WRlZRle\nS6lUPrYX0mCltLqYmJjw1ltv8fnnn+Pk5ER+fj7jxo3Dx8eHmzdvEhYW1uvjamtrmTBhwoCupSvN\ndfXqVb755huCgoKYO3duv//8XQNTg4ODuXDhAtHR0cycOZOKigq+/PJL3njjDb7//nvc3d3ZtGlT\nt8eWlZUZXTrr/PnzVFdXs2XLlucaI+Pp6YlOp6Ojo4OCgoIBXKEYaBLwCPGIgUxptba2cvToURoa\nGti6dWu/i1B/TqvVMnLkSBISEgzjAyIiIhg/fvyQzv0yNzfn9ddf73H7qFGjCAsLMwwBLSgo4Nat\nW+Tk5ODk5ERCQgKenp5kZmYyf/58rl69yu9+9zvDrClLS0v+8Y9/0NDQgF6vR6fT0dDQQEtLS48e\nNYMd8MDDkRJvvfUW33zzDRYWFnz44Yd0dnYaZiv19p4/bw3P4yiVSmbMmMGkSZOIj49nz549LFq0\niICAgH4HuNbW1ixdupQZM2Zw9uxZampq8PDw4KuvvgIenlKsrq7utpNkbAFPWloaWVlZbN26tV/F\n211p43v37lFSUkJJSQmtra386U9/Ij4+nt///veDuGrxvCTgEeIRA5XSKigoIDY2loCAANauXftM\njd/0ej1lZWXcuHGD27dv4+zsjKenJ/b29ixduvS51zhYLCwsmDx5Mnl5ebz22mv4+PiQl5fHvXv3\nyMrKoqWlhc7OTmbMmIGlpSWlpaVMmTKFbdu28Y9//IOWlhZ8fX25f/8+f/3rXxk/fjwhISF4eXmh\nUCiGJOCBhz1YurpPd3Z2Ym1tzahRo7h37x4eHh7d7qvRaGhpaWHkyJGDth4bGxtWrlxJaWlpt9Nc\nz9IDxsnJibfffpvCwkJiYmIMt3d2dhIfH8/GjRsNt5WVlQ37bLmBcvv2bRITE9m6dWu/mz02NTXx\n9ddfG34nAEaPHs0HH3zA7t27B2O5YgBJwCPEI543pdXZ2WnooLtq1apnOkreW13Orl27GDlyJE1N\nTfz9738nMjLyhe54W1FRQWFhIb/85S8xNzdn3LhxLFq0iI8//pjW1lby8vIwMTHBzs6OtLQ0PD09\nGTFiBFu3buXbb79lwYIFODk50dbWxs2bNw0nlkJCQoYk4NHr9cTGxjJjxgz0ej0HDhxg8+bN+Pj4\nUFBQ0CPgqa2tZdSoUUMyVNLd3Z2dO3ca0lyBgYHMmzfvmd4Tb29v7O3taWtrM9Rm5eXlUVxczLlz\n5/jmm2+or68nNDT0sd2jXxbFxcWcOHGCzZs3P1ONm52dHREREYbDAyYmJi/0Fw/RnUL/LKX/Qhip\nzs5O/uM//oN/+Zd/eaZamOrqamJiYnBwcGDZsmX9Ckh665cTFBTUa13OkSNHsLW1Zd68ef1e41DZ\nu3cvfn5+PfrzJCYmcunSJVasWIG3tzfXrl3j/PnzmJiYGJoc+vr64uLi0u3P3dUfKCMjg+vXrzNq\n1CgWLVrE+PHjByXIuHLlCpmZmWzbtg2lUsn+/fsNJ8kuXLjA9u3bu93/1q1bZGVl8eabbw74Wp6k\npaWFuLg4CgoKnjnNBQ9TpuXl5SQlJZGfn09cXBxXr16ltbUVACsrK44cOdKv00wvksrKSvbu3cu6\ndet6zDnrq9LSUg4cOIBKpaKxsRFXV1d27NgxwCsVg0V2eIT4ma76nf5eMPR6PWlpaSQkJLBw4UKC\ng4P79BxarZb8/Hxu3LhBYWEhPj4+farLiYiI4Msvv2TWrFn9biA3FAoKCmhoaDDMnvq5yspKQydr\nc3NzwsPDSU1N5Z133qGlpYW8vDwOHjxIZ2cnEyZMwM/PDy8vL0xMTBg7dixjx45Fr9ej0WhISEjg\n2LFjTJ06lalTpw5YOqmmpoYLFy6wbds2w+ewZs0avvjiC5ycnKiuru6R+hys+p2nsba27pbmSk9P\nJyoqqt9pLpVKhbu7Oxs2bCA7O5t/+7d/Q6PRGH7e2trK7373O9LT0wf6jzDoGhoa+P7774mKinqm\nYEev15OamkpSUhIrVqzAw8ODL7/8kkWLFg3CasVgkYBHiJ95lvqdBw8eEBsbS1tbG9u3b3/iwEvo\nvS4nMDCQFStW9Pm1HRwc8PT0JCMjo0/N0oaSXq8nLi6O+fPn9wjasrKyqKiowMfHh6ysLKZOnYpC\nocDb25u7d+8yffp0vLy8WLx4MbW1teTm5pKYmEhMTAxeXl74+voyYcIEQ8AUFBREVVUVGRkZfPrp\np7i6uhISEvLMPYfgYRB66NAhIiMjuwUw5ubmbNy4kS+++AIHBweKiorw9/c3/Ly2tvaZ0pcDpSvN\nlZ6e/txprrq6OiwtLbsFPPCwYebLprW1lX379hEREcHkyZP7/fiuWWcNDQ1s377dkAr78MMPB3qp\nYpBJwCPEz/S3fic3N5djx44REhLy1EGWT6rLeRYRERH8+OOPTJ8+fUhPaT3NzZs3UalUPZoINjY2\ncuLECTZu3EhzczOpqamGY+fe3t7k5OQYOhh3DcV0dHQkIiKC1tZW8vPzycvL4/Tp04bdMxcXF5yd\nnXnjjTdYuHAh2dnZpKWlceLECYKCgggJCen3rsuFCxcMg0UfNWrUKNauXcv+/fvJysrqFvDU1NQ8\ndrzGUFEqlUyfPp1JkyYRFxfHnj17WLhwIVOmTOnXrmVQUFCPmWlKpZK5c+cO9JIHVUdHB99//z3+\n/v6PbSXwJFVVVURHR+Pl5cWaNWsGZOK8GD7y6QnxM309kq7RaDh9+jQFBQWsX7++RwFrl77OsXoW\nrq6uODo6cuPGjcf2qxlqnZ2dnD9/nlWrVnX78+l0Og4fPkxYWBhjx45Fq9Vy9OhR6urqsLe3x8vL\ni5MnT6LT6Xqtx7GysiIoKIigoCC0Wi2ffvopGo2G/fv3o9fr8fX1xc/Pj0mTJhEYGEhNTQ0ZGRl8\n9dVXODo6EhISwqRJk556wSopKSEzM5P333//sZ+Pl5cXM2bMIDk5mWXLlmFubo5er6e2tva5uywP\nlEfTXF2nufpadGxjY8Mnn3xieB9UKhXW1tb8+c9/HuSVDxytVsvBgwdxcnJi/vz5/X789evXOXPm\nDIsXLzbagamvGpmWLgQPv52fOnWK4uJimpubgYcpjN6KjisqKti7dy9WVlZs3Lixxw6CVqslLy+P\nc+fOcerUKRQKBdOnT2fp0qX4+fkZjjoPBFtbW+Lj45k+ffqgNhzsq8uXL6PRaJg9e3a325OSkqir\nqzN0tFUqlTQ3N1NTU4OXlxfm5ubcuHGDsWPHPnWWllKpJD09nSVLlrBgwQJ8fHxobm4mPT2duLg4\nysvLMTMzY+rUqcyePRsLCwsyMzM5c+YMTU1N2Nra9trlur29nX379hEVFYWrq+sT1+Dl5UViYiKF\nhYVUVFTQ2tpKcXExc+bMeSE+hy52dnaEhITQ2dlJbGwszc3NuLm59WmnIigoiC1btjB+/Hg2btzI\n3/72t2GpUXoWer2eI0eOoNFoWLNmTb+K2js7Ozl+/Dg3btxg06ZN+Pj4DOJKxVCSU1pCAPfv3+dv\nf/ub4b8VCgWLFi0iMDAQvV5vmOadnJxMcnIyr7/+OlOmTDHc/+d1OVlZWTg5OREYGMikSZMGdRCp\nXq/n888/Z/bs2d3SK8NBrVbz17/+lS1btnQrmC0vL+e7775j165d3SaQV1ZWsn//fn7zm9+gUCg4\ndeoU1tbWvPbaa099rb/85S9s3ry5R71US0sL+fn55ObmUlRUhLOzs2H3R6VSkZmZSWZmJra2toSE\nhBAQEGAo+o6NjUWpVLJ8+fInvnZdXR0pKSlkZmYa0j4qlQqtVoupqSkzZsxg4cKFfX7fhkpLSwvx\n8fHk5+ezaNGifqe5XiZxcXEUFxfz7rvv9quov76+ngMHDjBq1ChWrFgxJL2exNCRlJYQPGzC5urq\nSnl5OfAwhTJ9+nS+/vprWltb2bRpE8eOHUOn07Fz505D3c3j5lgNZvO5n1MoFMyePZvExEQmTpw4\nrBewxMREfH19uwU7HR0dxMTEEBUV1S3YgYf1NxYWFhQXF+Pl5YW3tzepqal9CngeNzzU2tqa4OBg\ngoOD6ezspLi4mNzcXPbt24dKpcLX15eVK1fS0dFBZmYmZ8+eZdKkSdjb21NSUsL777//1Neuqanh\n6tWr3W7TarXAwwD0cenN4WZtbc2KFSsoKyvrdprrZe+t86jLly+Tk5PDtm3b+hXs5OXlceTIEWbP\nnk1YWJjRBoOvMklpCfF/rKysyM7ORqFQsHr1apqbm0lLS0OtVpOenk5wcLDh2//169c5deoUSUlJ\njBw5ksjISBYsWICnp+eg7uj0xtHRkZSUFJydnQd8YGhfNTU1cfToUdavX9/tW/Hx48exs7N7bBCj\n0WgoKChg4sSJ2NjYcPr0aWbOnPnEImy9Xs+5c+eIjIx8YqpCqVQa+vrMnDkTT09PGhsbuXr1KpmZ\nmTg6OhIUFIRarSYpKQlLS0tMTU1xcHB4Ysqn6+elpaXodDpMTEzQ6XQoFAqCgoKYNWtWH96x4WNr\na8vUqVPR6XT9TnO96G7fvs2FCxfYsmULI0aM6NNjdDod586dIzExkTfffJPJkydLsGOkXv7fcCEG\niJ+fHwqFAltbW3x9fdmzZ4/hWG5XyuLgwYP96pczFBQKBRERESQmJj5zQ7Xndf78eUJCQrrV32Rl\nZVFcXPzEXZPAwED++te/Gjonu7i4UFJS8sS6ic7OTlQqVb/ed4VCgYuLCy4uLsydO5fm5mZD6is/\nPx9bW1s8PT3Jzc0lPj6eiRMnEhISgru7e68Xv4iICBoaGkhPT8fJyYmKigqAHrVLLyqlUkloaGiP\n01yBgYEv7cW+qKiIkydPsnnz5j7vsD548ICYmBgUCgW7du3q96gJ8XKRHR4h/k9VVRVXrlxh8uTJ\n1NTUcPv2bUOBrU6no7q6mtDQUJYvX05QUBCOjo5DMkagL5ydnTl//jzjxo3r8zfbgVJdXc358+dZ\nt26dYZegqamJ/fv3s379+if2JTIzM6OsrAy9Xs+YMWNobGyktrb2iQFPa2srmZmZz7WTYm5uzpgx\nY2htbaW1tZUFCxZQV1dHaWmp4dRVRkYG6enpaLVaHBwcMDU1NTxeoVAwfvx4cnJyGD16NA8ePMDf\n35/09HQCAwNfmN+LpzE1NcXPzw8PDw8uXLjAzZs3cXV17bWo+0XWVQ+2fv36Pg85LSkpMXQDl3qd\nV4Ps8AjBw8LagIAANBoNOp2OjRs34u3tjZmZGb6+vkyfPp0xY8YM9zIfS6VSER4eTlJSEuvXrx/S\n146PjyciIsKQytPr9fz000/MmDGjTxefoKAgLl++zNSpU/H29ubUqVNPvP9AzdGqrq4mISHB0Cxy\n4sSJ6PV6KioqyMvLo6Ojg7q6OtLS0jh37hzjx483NEZUKBSUlpYSHR1tmKs0bdo0NmzYwNGjR3sc\ny3/Rubm5sWPHDjIyMvj2228JCAggMjJyyNOzz6K+vp7vv/+epUuX4unp+dT76/V6UlJSSE5OZuXK\nlUyYMGHwFyleCBLwCAGcOnWK9vZ2w9ygCxcu8K//+q8D0i9nqISEhHDp0iVqamqGrB/M3bt3qa6u\n7hZkJScno9Pp+pze8fX15dixY9TX1zN27Fjq6upobW197ByygQh4tFotP/30EwsWLOi2A6VQKHB1\ndcXV1ZV58+bR1NREXl4e2dnZ5OfnU1hYiEqlwt/fn507d3L//n3D1OzLly+Tm5vL73//e5KTk4mI\niHiuNQ61n6e54uPjX4o0V0tLC/v27eO1117r0eiyN2q1miNHjtDY2MiOHTuG7HCBeDG8HPuuQgwy\nHx8fw//UzczMmDlzJmPHjn1h/0ffGzMzM6ZPn05ycvKQvJ5er+fs2bNERkYaUlnl5eUkJyezevXq\nPqd1VCoVAQEBXL9+HZVKhYeHB0VFRY+9/+NOaPXH+fPnsbOze2rDRltbW0JDQ9m8eTO/+93vWL16\nNe7u7uzdu5f6+npDsAMP34/29nY0Gg2XL18mPz+f9vZ2srKyuHLlynOtdyhZWVmxfPlyNmzYQFpa\nGl999RWVlZXDvaweurooT5482dCh+0mqqqr47LPPsLa2ZuvWrRLsvIIk4BECmDt3Lv/1X/9l6IS8\nZ8+e4V7SM5kxYwbZ2dk0NTUN+mtlZ2ej1WoN/Yg6Ojo4dOgQb7zxRr8vJsHBwVy/fh29Xo+3tzeF\nhYWPva9arX6uHZ7i4mKuX79uaILYV2ZmZkycOJGNGzcyefJkOjo6etyntbWV3Nxc3Nzc2L9/P3/4\nwx/46aefDGmvl8nYsWPZvn07gYGB7Nu3j5MnT6JWq4d7WcDDHbro6GhGjx5NZGTkU++fmZnJt99+\ny9y5c1m6dKlRnEgT/ScBjxD/Z/fu3dy6dYsffvihR8+Yl0XXCIaUlJRBfR2tVkt8fDwLFy40BA2n\nT5/Gzc2NgICAfj+fi4sL5ubm3L17F29v76fu8DxrwKNWqzl8+DArVqx4rhM5U6ZM6bU43NramnHj\nxpGdnY1Op0On0xlOld28eZPq6mpDz56XQVea68MPP6Szs5M9e/aQmZnJcPar7eqirFKpWLZs2ROD\n1s7OTo4cOUJiYiJbtmwhMDBwCFcqXjQS5gphZMLDw/nkk0+YM2dOn+aCPYuMjAxGjhxpOE2VnZ1N\nUVFRnxr39aarh01mZiYrV65Eo9FQX1/fa1+h5wl4Tp48yYQJE567UHX58uU4OzujVqsNrQuUSiVm\nZmZ89NFHtLe3891339HQ0IBer8fW1pacnBwuXLhAU1MTjo6OjB49uts/L/KR6K4017179wyzuaKi\nonBxcRnytcTFxVFXV8e77777xLRpfX090dHRODg4sHPnTjmFJeRYuhDGxsLCgtraWmpra/t0aqW/\n2tvbOXDgACtXrmTEiBF9PoL+NKNGjeLEiROEhYVx//599Hp9rzOt7ty5g0ql6nfPoVu3bnHz5k3e\neuut5+6dpFKp2LhxI6WlpeTl5aFSqVi6dCmvv/46GRkZjBgxgoULF9LW1kZVVRWjR4/mrbfeIiws\njJkzZxoa/dXW1pKVlcX58+dJTU2loKCAqqoqHjx4ADz8LF+kI+5dTQu7dlkaGxtxd3cfshRRamoq\nt27dYvPmzU+s48rNzeWHH34gLCyMRYsWSQpLADJLSwijVFNTw9dff82vf/3rfrXX74sLFy5QV1fH\nmjVr0Ov17N27l3HjxjF37tznfu4ffvjBMBPszp07rFu3rsd9Tp8+zYgRI/rVh6exsZFPP/2UTZs2\nPXUw6PO4fPmy4Vi9SqXCzc2NyZMnk5qaSkREBCEhIb0+Tq/X09zcTFVVVbd/una5XFxccHZ2NuwG\njRgxYtgL6ltbW4mPjycvL48FCxYQFBQ0qGu6desWZ8+eZdu2bY9NOXd1Tb558ybr1q3D3d190NYj\nXj4S9gphhBwdHfHw8ODatWuEhYUN2PM+ePCAtLQ0du3aBUBKSgqdnZ19mn/VF8HBwVy+fJnVq1dz\n5swZ9Hp9j4toe3t7v47d6/V6YmNjCQsLG9RgBx52jj59+jR6vR6tVktJSQmzZ89m48aNfP3114bP\n5VFdHb5tbW27pds6Ozu5f/++IQBKTU2lsrISvV7P6NGjuwVBzs7O3ZojDrZH01xds7kGo19VYWEh\np06dYvPmzY8Ndrq6JiuVSumaLHolAY8QRioiIoLo6GhCQ0MHbPzFhQsXCAoKYuTIkVRUVJCUlMTO\nnTsHLO3S1ZNHp9NhZWVFZWVljwtof2t4UlNT6ezsHJKxD5aWljg7O1NVVQWAp6cn48ePB2DVqlUc\nOHCA7du39/kUm4mJCWPGjOnxHjx48MAQBN29e5e0tDRqa2uxs7PrURtkZ2c3qDsvY8eONTQt/CaC\nnw0AABIXSURBVO6775g0aRKRkZEDVj9WUVFBTEwMb7755mMHnd69e5eYmBimTp3K3LlzX6g0oHhx\nSA2PEEbK1taW/Px8wxyp51VTU8OZM2cMTQb37t3LwoULGTdu3HM/dxelUklTUxO1tbXY2NjQ1tbW\nY0fk2rVrjBs3DgcHh6c+X1VVFUePHmXTpk2PbWQ40NRqNaWlpSxYsICMjAza29vx8fHB3t4ehUJB\nfHw8gYGBzxWEmpmZYW9vj7u7O/7+/oSGhhIREYG3tzeWlpY0NTWRn59PYmIiCQkJ5OXlUV5eTlNT\nE1qtFgsLiwGta+lq2BgcHGzYjbG0tMTFxeW5gq36+nr27t3L0qVLDYHjz3V1TT59+jQrV65k2rRp\nw57qEy8uqeERwogVFhZy8uRJPvzww+e+EERHR+Pq6srs2bM5duwYGo2G1atXD9BK/7+Kigqio6NZ\nvHgx6enpvPPOO91+/sUXX7B48eKn1md0dnby2WefER4eTnBw8ICv80mv29LSgp2dHTdv3uTQoUNE\nRkYyZ84cQ7Fve3s769evH5KLc2trK9XV1VRVVVFZWUl1dTX379/HxsamR1qsKyh7XuXl5Zw4cQKF\nQvHMaa6Wlha+/PJLwsPDCQ0N7fFztVpNbGwsTU1NrF+/XhoJiqeSlJYQRszLywtTU1Nyc3OZOHHi\nMz9PaWkp9+7dY/Xq1eTk5FBQUMAHH3wwgCv9/1xcXDAzM0OlUlFaWkpnZ2e33Yi+Nh6Mj4/HwcGB\noKCgQVnn45iYmBjqTKZMmYJarebEiRNYWVkRGhrK0qVL+eabb7h48SLz5s0b9PVYWVnh6enZ7cSe\nTqejrq7OkBa7fv06VVVVtLa2dguAugKi/qanXF1d2b59O9euXeO7777D39+f+fPnG55Hq9UaBvP2\npqOjg++++46AgIBeg53KykoOHDiAj48Pa9eulVNYok/kt0QII6ZQKJg9ezaJiYn4+fk907d3vV5P\nXFwc8+bNQ61Wc+zYMd56661B62uiUCgIDg4mOzsbJycnSktLux1B70sNT2FhIbdv3+aDDz4Y9hTH\n9OnTaWtr4/jx41haWjJ58mTeeustPv/8c5ydnfs0A2qgKZVKHB0dcXR0ZPLkyYbb1Wq1YTeoqqrK\n0CzRwsKiR22Qg4PDE2tlFAoFISEh+Pv7G2ZzLViwgODgYPbt24eZmRkbN27s8biuLspjxozpNSC8\ndu0acXFxvP7664Yu30L0hQQ8Qhi5iRMncu7cOe7evftMfXny8vJQq9UEBgby/fffExoaOujHfadM\nmcKePXsICQmhsLCwXwFPW1sbsbGxrFixYsjqdp5mzpw5tLW1ERMTg7m5OePHj+ett95i37592Nvb\nD0sDv95YWFjg4eHRrW5Kr9dTX19vCIK6+gY1NTXh5ORk2AVycXFh9OjRPd5zS0tLli1bRkhICCdO\nnCAxMZGmpiYUCgW5ubn4+fl1e63Y2FhMTExYunRpt2BVo9Fw8uRJSktLee+993Bychr8N0QYFQl4\nhDBySqWSWbNmkZiY2O+AR6fTERcXx6JFi0hLS0Oj0TBnzpzBWejP2NjY4OHhgUKh6DZmQqfTodFo\nHhvw6PV6jh8/zsSJE3stch1OS5Ysoa2tje+//55t27bh5uZGVFQU+/fvZ+fOnS/sMWqFQoG9vT32\n9vaGHknwMO30892gnJwcqqqqMDU17bEb5OjoiKurK++++y5//OMf6ezsBODw4cN89NFHhs/z7Nmz\nNDQ0sHnz5m67R3V1dRw4cAAHBwd27NghXZPFM5GAR4hXQGBgIBcuXKCioqJfBaSZmZlYW1szYsQI\nYmNj2bFjx5Ad+Q0KCiItLY379+/T1taGpaUlHR0dmJmZPTZN1ZWC2blz55Cssb9WrVpFW1sbX331\nFe+//z6TJ0+murqaH3/8kS1btgxY+4ChYGZmhpubG25ubobb9Ho9TU1NhgLpvLw8Ll26RENDgyEF\n1tHRgVKpRKfToVar+fHHH1mzZg3Xrl3jzp07bN26tVs/oZycHI4ePcrcuXOZPn36sKcoxctLAh4h\nXgEmJiaEh4eTlJTUa/fi3mg0Gi5cuMDatWs5dOgQS5Ys6XW21WDp6snj4uJCcXEx/v7+T0xnNTQ0\ncPr0ad55550hbcDXXxs3buSrr77is88+Y/fu3cybN4/q6mqOHz/e7wnuLxqFQoGdnR12dnb4+voa\nbtdoNNy/f5+7d+9ibW1NfX09jY2N6PV6srOzsbe3Z9SoUeTl5RkKm3U6HfHx8dy+fZuNGzd2C6yE\neBbSnUmIV0RISAhFRUXU1tb26f6pqam4u7tz69YtxowZM+STpk1MTAgICMDExITCwkLg8Se0dDod\nhw8fJjw8fFA6/Q60LVu24ODgwCeffEJbWxurV6/m3r17pKWlDffSBoWpqSmurq6Eh4ezadMmfvnL\nX/LP//zP/OY3v2Hx4sXAw/qhrtNtDx484Ntvv6Wqqopdu3ZJsCMGhAQ8QrwizM3NCQ0NJTk5+an3\nbW1tJSUlBS8vL+7cuUNUVNQQrLCn4OBg7t+/bwh4HrfDk5KSAtCv+VrDqWv8gaWlJXv27AEe7vwk\nJiZSUFAwzKsbGgqFghEjRhAZGUllZSW5ubkAFBcX8+mnn+Lp6cnbb7/9whSei5efBDxCvELCwsLI\nysqiubn5ifdLSEjA19eXCxcusHr16idOph5MY8aMwdLSkpaWFhobG2lvb++xloqKCpKTk1m1atVL\nNVJAqVSye/dulEole/bswcbGhrVr1/LTTz/1eRfOWNjb22Nubk5SUhIHDx5kxYoVzJs376X6PMWL\nT0ZLCPEKMTU1pbm5mZKSEurq6sjJycHb2xt4+M06NzcXCwsLTp48iampKf7+/kPapfhRCoUCjUZj\n6BCclZWFSqXCy8sLhUKBXq9n3759zJ8/f0BHXAwVpVJJSEgIycnJ3Lhxg4ULF2Jubs7p06cJDAx8\nZRrqqdVqYmJiKC8vZ/PmzYM+5FW8mmS0hBCvELVazcWLF0lNTUWlUmFra8uvf/1rAM6cOUNqaqqh\nKZ2pqSlbt24d1m/ZRUVFxMTE0NLSYrjNzc0NvV5PQ0MDHh4eKJVK1q5d+1IX+zY1NfHxxx/j6OjI\njh07OHXqFA0NDWzYsMHodzkqKyuJjo5m/PjxLFmy5KU6qSZeLsb9N0kI0U1sbCyXL18GHna0/fnF\npaOjA71ej1arpaqqio6OjmEPIi5evNgt2AEYOXIktbW1tLS0kJ2djbW1NTqdbphWODBsbW354IMP\nqK6uZt++fSxZsgSNRkN8fPxwL21QZWRksHfvXubPn09UVJQEO2JQScAjxCtkxYoVuLu7G1IlP989\naG9vN/y7SqVi5syZwx7wbNiwAUdHx2632dnZdVtrWlqaUZxusre3Z/v27dy9e5dDhw6xfv16srOz\nuX79+nAvbcBpNBpiY2NJSUnhvffeIyAgYLiXJF4BEvAI8QqxtLRky5YtzJgxA+ge5DQ0NAAPT3P9\n4he/YOrUqcOyxp+zsLBg165dODs7Aw9rerRareHfLSwsWL58OWFhYcO5zAEzZswYNm/eTHZ2NufO\nnWPDhg2cOXOGsrKy4V7agKmrq+OLL76gs7OTnTt3yogIMWSkhkeIV9Tly5e5ePEiDx48MBz1VigU\nfPTRRy/cUWCtVsuf//xnKisrCQgI4O7du8ydO5eIiIgXusngs8rNzeXHH38kIiICNzc3jh8/zo4d\nO7C1tR3upT0X6ZoshtOrcQRACNHDkSNH+POf/0x7ezsqlQq9Xs9///d/v3DBDjxMsV25coXDhw9j\nYmLCuXPnjGZXpzd+fn6sXLmSw4cPs2jRImbMmMGPP/7Ie++991IGeF0z2bKysqRrshg2ssMjxCso\nKyuL0NBQ2traut1uaWlJZWXlC7eToFaruxUn//a3v+WPf/zjMK9q8KWmpnL69GlWrFhBUVERer2e\nNWvWvFQ7I83NzRw8eBAzMzNWr179QgbU4tUgNTxCvIJOnDhhmFj9c6ampiQkJAzDip7M3NycCRMm\nYGVlhZWVFa+99tpwL2lIzJw5kzlz5nD06FF8fX2pq6sjKSlpuJfVZ8XFxXz22Wd4e3tL12Qx7CSl\nJcQryNraGhMTEzQaTbfb9Xo9NjY2w7Sqx1MoFKSkpHDw4EEmTJjAvHnzhntJQyYyMpK2tjYOHTrE\nmjVrOHPmDE5OTvj5+Q330h5Lr9eTlJREamoqq1evxsfHZ7iXJISktIR4Fd2/fx9PT09aW1u73T56\n9Gju3bsn/VBeQDExMWRlZbF8+XLOnj3Lli1bDKfXXiRtbW3ExsbS0tLCunXrDANBhRhuktIS4hXk\n5OTETz/9hJ2dHba2ttjY2ODm5kZcXJwEOy+otWvX4uXlxbFjx5g1axb79+/vEbAOt4qKCj777DPs\n7Ox47733JNgRLxTZ4RHiFdbR0UFaWhrm5uZMmzbN6McYGIPPP/+c6upqgoKCqKmp4Z133hn2IFWv\n15ORkcG5c+d44403pJGgeCFJwCOEEC8RnU7H3//+d5qbm3F1dcXR0ZGoqKhhW49Go+H48eOUl5fz\n5ptv9uiMLcSLQgIeIYR4yXR2dvLxxx+j0WiwsLAgPDyc0NDQIV9HbW0tBw4cwNnZmWXLlmFmZjbk\naxCiryTgEUKIl1BHRwd/+ctfUKlUaLVa1q9fj6en55C9fnZ2NseOHWPevHmEhoa+VL2BxKtJAh4h\nhHhJtba28r//+79YWFig0WjYsWMHo0aNGtTX1Gq1xMXFkZ2dzfr16xk7duygvp4QA0UCHiGEeIk1\nNTXx8ccfY21tjZmZGdu3bx+01JJ0TRYvMwl4hBDiJVdTU8Mnn3yCtbU1rq6uvPnmmwOeYioqKuLQ\noUNMnz6d1157TVJY4qUjAY8QQhiBe/fu8cUXX2BpaUloaCiRkZED8rx6vZ7ExETS0tJYvXo13t7e\nA/K8Qgw1CXiEEMJIFBYWsm/fPkxNTVmxYgWTJ09+rudra2vj8OHDtLW1sW7duhduqKwQ/SEBjxBC\nGJGsrCwOHDiAiYkJ27ZtY8yYMc/0POXl5Rw4cAA/Pz8WLVo07M0NhXheEvAIIYSRycjI4OjRo1hY\nWLB79+5+DYT9edfkqKio594lEuJFIQGPEEIYoaSkJOLi4hg5ciSTJ0+mpqaGDRs2PPExGo2GY8eO\nUVFRIV2ThdExGe4FCCGEGHgRERG0tLSQkpJCcnIySqWS9vZ2zM3Ne71/bW0t0dHRuLi4sGPHDuma\nLIyOBDxCCGGEdDodTU1NKBQK9Ho9er2ewsJC/P39e9w3KyuL48ePExkZybRp0+TIuTBKEvAIIYQR\nUqvVlJSUYGJigkajQafTcenSpW4BT1fX5JycHDZt2oSrq+swrliIwSU1PEIIYaT0ej0lJSVcvXqV\nW7duARAVFYVSqcTHx4dDhw5hYWHB6tWrsbS0HObVCjG4JOARQohXQHt7O//0T//Ep59+iomJCT4+\nPvztb39j9uzZksISrwRJaQkhxCvA3Nyc4uJiNBoNGo2GrKwspk+fLsGOeGUoh3sBQgghhsbcuXOx\nsrLCzMwMLy+vx57YEsIYSUpLCCFeEXq9nh9//JGysjK2bNmCk5PTcC9JiCEjAY8QQgghjJ6ktIQQ\nQghh9CTgEUIIIYTRk4BHCCGEEEZPAh4hhBBCGD0JeIQQQghh9CTgEUIIIYTRk4BHCCGEEEZPAh4h\nhBBCGD0JeIQQQghh9CTgEUIIIYTRk4BHCCGEEEZPAh4hhBBCGD0JeIQQQghh9CTgEUIIIYTRk4BH\nCCGEEEZPAh4hhBBCGD0JeIQQQghh9CTgEUIIIYTRk4BHCCGEEEZPAh4hhBBCGD0JeIQQQghh9CTg\nEUIIIYTRk4BHCCGEEEZPAh4hhBBCGD0JeIQQQghh9CTgEUIIIYTRk4BHCCGEEEZPAh4hhBBCGD0J\neIQQQghh9CTgEUIIIYTRk4BHCCGEEEZPAh4hhBBCGD0JeIQQQghh9CTgEUIIIYTRk4BHCCGEEEZP\nAh4hhBBCGD0JeIQQQghh9CTgEUIIIYTRk4BHCCGEEEZPAh4hhBBCGD0JeIQQQghh9CTgEUIIIYTR\nk4BHCCGEEEZPAh4hhBBCGD0JeIQQQghh9CTgEUIIIYTRk4BHCCGEEEZPAh4hhBBCGD0JeIQQQghh\n9CTgEUIIIYTRk4BHCCGEEEbv/wE3vexcBUdelQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Here's another example with a random graph\n",
    "\n",
    "n = 100\n",
    "# G = nx.fast_gnp_random_graph(n,1/n,directed=True)\n",
    "G = nx.erdos_renyi_graph(n,2.5/n, directed=True) \n",
    "\n",
    "# Find the ranking of all nodes, and the max ranged node\n",
    "\n",
    "vals = np.array(list(nx.pagerank(G).values()))\n",
    "m = np.argmax(vals)\n",
    "print(max(vals),m)\n",
    "\n",
    "# Draw the network\n",
    "\n",
    "fig,ax = plt.subplots(1,1,figsize=(10,10))\n",
    "\n",
    "nx.draw(G, \n",
    "        with_labels=False, \n",
    "        node_color=['black' if k != m else 'orange' for k in range(n)],\n",
    "        edge_color='gray',\n",
    "        node_size=(500*vals)**2,\n",
    "        node_shape='o',\n",
    "        pos=nx.kamada_kawai_layout(G))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 633,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-22T01:23:21.680982Z",
     "start_time": "2019-11-22T01:23:21.518079Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAAD8CAYAAABXXhlaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Show the stochastic matrix\n",
    "\n",
    "plt.imshow(nx.google_matrix(G));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 672,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-22T01:34:20.490773Z",
     "start_time": "2019-11-22T01:34:20.470156Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([38, 97, 40, 42, 66, 83, 10, 52, 86, 24, 70, 54, 37, 59, 34,  4, 39,\n",
       "       98, 41,  0,  1, 79, 71, 73, 85, 20, 90, 68, 17, 33, 63, 18, 72,  9,\n",
       "       67, 55, 13, 31, 35, 27, 36, 30, 76, 47, 87, 69, 94, 57, 46,  3,  6,\n",
       "       53, 23, 75,  7,  8, 96, 32, 74, 28, 62, 65, 91, 78, 95, 44, 93, 12,\n",
       "       16, 21, 15, 80, 92, 11, 49, 50, 60, 45, 51, 99, 77, 84, 82, 61, 89,\n",
       "       25, 14, 58, 43,  5, 48, 26, 88, 19, 64, 29, 81, 22, 56,  2])"
      ]
     },
     "execution_count": 672,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vals = np.array(list(nx.pagerank(G).values()))\n",
    "ranks = vals.argsort()\n",
    "ranks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 673,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-22T01:34:20.859676Z",
     "start_time": "2019-11-22T01:34:20.667408Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x12d16ef98>]"
      ]
     },
     "execution_count": 673,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX0AAAD8CAYAAACb4nSYAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(vals,'o')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}