{
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"\n",
"\n",
"\n",
"# One-dimensional stationary heat equation\n",
"\n",
"### Examples - Thermodynamics\n",
"\n",
"By Andreas S. Krogen, Jonas Tjemsland and Jon Andreas Støvneng.\n",
"\n",
"Last edited: March 16th 2018 \n",
"\n",
"This notebook is based on an assignment given in the course TMA4320 Introduction to Scientific Computation at NTNU, spring 2017.\n",
"___"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"## Introduction\n",
"\n",
"The heat equation is a partial differential equation that describes the distribution of heat in a system.\n",
"It can be solved in order to find the temperature variations in a physical body.\n",
"In this notebook, we shall first derive the heat equation describing a system containing a heat source.\n",
"We will be focusing on the steady state solution, when the system has stabilised and there is no time dependence.\n",
"As a concrete example, we will be looking at a thin rod with fixed temperatures at the ends. The rod is thin enough to be regarded as one-dimensional and is heated by a Gaussian heat source.\n",
"The stationary heat equation will be solved using a collocation method, approximating the solution as a sum of Lagrange polynomials."
]
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"## A non-rigorous derivation of the heat equation\n",
"Consider a uniform system of fixed volume $\\mathcal{V}$ and no exchange of particles with the environment.\n",
"\n",
"Let $u(\\mathbf{r}, t)$ denote the energy density of the system, $\\mathbf{j}(\\mathbf{r}, t)$ the heat flux going out of the system into the environment and let $\\dot{q}(\\mathbf{r}, t)$ denote the heat that enters the system from the heat source per unit volume per unit time (heating rate per unit volume). The change in the total energy of the system must be equal to the energy entering the system from the heat source minus the energy that flows out of the system, hence\n",
"\n",
"\\begin{align*}\n",
"\\frac{\\mathrm{d}}{\\mathrm{d}t} \\int_\\mathcal{V} u(\\mathbf{r}, t) \\,\\mathrm{d}V &= \\int_\\mathcal{V} \\dot{q}(\\mathbf{r}, t) \\,\\mathrm{d}V - \\oint_{\\partial\\mathcal{V}} \\mathbf{j}(\\mathbf{r}, t) \\cdot \\mathrm{d}\\mathbf{S} \\\\\n",
"\\int_\\mathcal{V} \\frac{\\mathrm{\\partial}}{\\mathrm{\\partial}t} u(\\mathbf{r}, t) \\,\\mathrm{d}V &= \\int_\\mathcal{V} \\dot{q}(\\mathbf{r}, t) \\,\\mathrm{d}V - \\int_\\mathcal{V} \\nabla\\cdot \\mathbf{j}(\\mathbf{r}, t) \\,\\mathrm{d}V \\\\\n",
"\\int_\\mathcal{V} \\left[ \\frac{\\mathrm{\\partial}}{\\mathrm{\\partial}t} u(\\mathbf{r}, t) + \\nabla\\cdot \\mathbf{j}(\\mathbf{r}, t) - \\dot{q}(\\mathbf{r}, t) \\right] \\,\\mathrm{d}V &= 0\n",
"\\end{align*}\n",
"\n",
"$$\\implies \\frac{\\mathrm{\\partial}}{\\mathrm{\\partial}t} u(\\mathbf{r}, t) + \\nabla\\cdot \\mathbf{j}(\\mathbf{r}, t) - \\dot{q}(\\mathbf{r}, t) = 0 \\quad .$$\n",
"\n",
"We have assumed that the volume of the system is constant, which implies that there is no pressure-volume work involved.\n",
"The first law of thermodynamics then reads\n",
"\n",
"\\begin{align*}\n",
"\\mathrm{d}U &= \\mathrm{d}Q \\\\\n",
" &= C_V \\mathrm{d}T \\quad ,\n",
"\\end{align*}\n",
"\n",
"where $U$ is the internal energy, $T$ the temperature of the system and $C_V$ the heat capacity of the system under constant volume.\n",
"The partial time derivative of the internal energy density $u$ can thus be expressed as\n",
"\n",
"$$\\frac{\\mathrm{\\partial}u}{\\mathrm{\\partial}t} = \\rho c_V \\frac{\\mathrm{\\partial}T}{\\mathrm{\\partial}t} \\quad .$$\n",
"\n",
"Here, $c_V$ is the heat capacity per unit mass and $\\rho$ is the mass density of the system.\n",
"\n",
"Fourier's law is an empirical law that states that the heat flux is proportional to the negative temperature gradient (heat flows from regions of higher temperature to regions of lower temperature),\n",
"$$ \\mathbf{j} = - \\kappa \\, \\nabla T \\quad .$$\n",
"\n",
"The constant $\\kappa$ is called the *thermal conductivity*.\n",
"\n",
"Inserting these two expressions into the energy conservation equation yields the heat equation with a source term\n",
"\n",
"\n",
"$$\\frac{\\mathrm{\\partial}}{\\mathrm{\\partial}t} T(\\mathbf{r}, t) = \\frac{\\kappa}{\\rho c_V} \\nabla^2 T(\\mathbf{r}, t) + \\frac{1}{\\rho c_V} \\, \\dot{q}(\\mathbf{r}, t) \\quad .$$\n",
"\n",
"We are interested in the stationary solution and no explicit time dependence implies that\n",
"\n",
"\\begin{align*}\n",
"0 &= \\frac{\\kappa}{\\rho c_V} \\nabla^2 T(\\mathbf{r}) + \\frac{1}{\\rho c_V} \\, \\dot{q}(\\mathbf{r}) \\\\\n",
"-\\nabla^2 T(\\mathbf{r}) &= \\frac{1}{\\kappa} \\, \\dot{q}(\\mathbf{r}) \\quad .\n",
"\\end{align*}\n",
"\n",
"In one dimension this becomes\n",
"\n",
"$$-\\frac{\\mathrm{d}^2}{\\mathrm{d}x^2} T(x) = \\frac{1}{\\kappa} \\, \\dot{q}(x) \\quad .$$\n",
"\n",
"The SI units of the relevant physical quantities can be found using dimensional analysis:\n",
"\n",
"We know that $[T] = \\mathrm{K}$ and $[\\dot{q}] = \\mathrm{Jm^{-3}s^{-1}}$, and wish to find the unit of $\\kappa$.\n",
"From the one-dimensional heat equation we have that\n",
"\n",
"\\begin{align*}\n",
"[\\kappa] &= [\\dot{q}(x)]\\, \\cdot \\left[\\frac{\\mathrm{d}^2}{\\mathrm{d}x^2} T(x)\\right]^{-1} \\\\\n",
" &= \\mathrm{Jm^{-3}s^{-1}} \\cdot (\\mathrm{K m^{-2}})^{-1} \\\\\n",
" &= \\mathrm{JK^{-1}m^{-1}s^{-1}} \\quad .\n",
"\\end{align*}"
]
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"cell_type": "markdown",
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"## Example: Modelling a thin metal rod that is heated by a candle\n",
"Consider a thin metal rod of length $L = b - a\\,$ and cross-sectional area $A$, which are both assumed to be constant.\n",
"The rod is heated by a burning candle that is modelled by a Gaussian heat source\n",
"\n",
"$$\n",
"\\dot{q}(x) = \\dot{q}_0 \\exp\\left[-\\frac{1}{2}\\left(\\frac{x-x_0}{\\sigma}\\right)^2\\right] \\quad ,\n",
"$$\n",
"\n",
"where $\\dot{q}_0$ is related to the power of the heat source $P_\\mathrm{src}$, $A$ and $\\sigma$, which will be explained later in this notebook.\n",
"A pair of heat reservoirs of constant temperatures $T(a)$ and $T(b)$ keep the temperature of the rod fixed at the ends. See Figure 1.\n",
"\n",
"**Figure 1:** A sketch depicting the physical situation. The ends of the metal rod are attached to heat reservoirs of temperatures $T(a)$ and $T(b)$.\n",
"\n",
"\n",
"We start by rewriting the heat equation into a more numerically tractable form\n",
"\n",
"\\begin{align*}\n",
"-\\frac{\\mathrm{d}^2}{\\mathrm{d}x^2} T(x) &= \\frac{\\dot{q}_0}{\\kappa} \\exp\\left[-\\frac{1}{2}\\left(\\frac{x-x_0}{\\sigma}\\right)^2\\right] \\\\\n",
"-\\frac{\\mathrm{d}^2}{\\mathrm{d}x^2} \\left[\\frac{\\kappa \\, T(x)}{\\dot{q}_0}\\right] &= \\exp\\left[-\\frac{1}{2}\\left(\\frac{x-x_0}{\\sigma}\\right)^2\\right] \\quad .\n",
"\\end{align*}\n",
"\n",
"We also introduce the dimensionless variables $\\tilde{x} = (x-a)~/~L$ and $\\tilde{\\sigma} = \\sigma~/~L$\n",
"mapping $a \\mapsto 0$ and $b \\mapsto 1$.\n",
"This yields the following dimensionless equation\n",
"\n",
"\\begin{align*}\n",
"-\\frac{\\mathrm{d}^2}{\\mathrm{d}\\tilde{x}^2} \\left[\\frac{\\kappa \\, T(\\tilde{x})}{L^2 \\, \\dot{q_0}}\\right] &= \\exp\\left[-\\frac{1}{2}\\left(\\frac{\\tilde{x}-\\tilde{x}_0}{\\tilde{\\sigma}}\\right)^2\\right] \\\\\n",
"-\\frac{\\mathrm{d}^2}{\\mathrm{d}\\tilde{x}^2} \\mathcal{T}(\\tilde{x}) &= f(\\tilde{x} ; \\tilde{x}_0, \\tilde{\\sigma}) \\quad .\n",
"\\end{align*}\n"
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"### Expanding the solution in a basis of Lagrange polynomials\n",
"The dimensionless heat equation can be solved by discretising the interval $[0,1]$ on the $\\tilde{x}$-axis and approximating the solution on the whole domain as a sum of Lagrange polynomials, demanding that the approximation satisfies the heat equation at each discrete point. The points on the domain, at which the approximate solution satisfies the heat equation, are called *collocation points*. For a short introduction to Lagrange interpolation, please have a look at the relevant section in our notebook on [polynomial interpolation](https://nbviewer.jupyter.org/urls/www.numfys.net/media/notebooks/polynomial_interpolation.ipynb).\n",
"The interval $[0,1]$ is discretised into $N$ points, $0 = \\tilde{x}_0 \\lt \\tilde{x}_1 \\lt \\dots \\lt \\tilde{x}_{N-2} \\lt \\tilde{x}_{N-1} = 1$.\n",
"The solution is approximated by\n",
"\n",
"$$\n",
"\\mathcal{T}(\\tilde{x}) = \\sum_{i = 0}^{N-1} c_i \\mathrm{L}_i(\\tilde{x}) \\quad ,\n",
"$$\n",
"\n",
"where \n",
"$$\n",
"\\mathrm{L}_i(\\tilde{x}) \\equiv \\prod_{k = 0}^{N-1} \\frac{\\tilde{x} - \\tilde{x}_k}{\\tilde{x}_i - \\tilde{x}_k}\n",
"$$\n",
"\n",
"is the Lagrange polynomial corresponding to the point $\\tilde{x}_i$.\n",
"The Lagrange polynomials have the important property $\\mathrm{L}_i(\\tilde{x}_j) = \\delta_{ij}$.\n",
"Inserting the polynomial expansion into the heat equation and applying this property yields \n",
"\n",
"\\begin{cases}\n",
"c_0 &= \\mathcal{T}_a \\\\\n",
"-\\sum_{j = 0}^{N-1} c_j \\mathrm{L}_j''(\\tilde{x}_i) &= f(\\tilde{x}_i), & i = 1, \\cdots, N-2 \\\\\n",
"c_{N-1} &= \\mathcal{T}_b\n",
"\\end{cases}\n",
"\n",
"This system of linear equations can be written in matrix form $A\\vec{c} = \\vec{f}$ :\n",
"\n",
"$$\n",
"\\begin{pmatrix}\n",
"1 & 0 & \\cdots & 0 \\\\\n",
"-\\mathrm{L}_0''(\\tilde{x}_1) & -\\mathrm{L}_1''(\\tilde{x}_1) & \\cdots & -\\mathrm{L}_{N-1}''(\\tilde{x}_1) \\\\\n",
"-\\mathrm{L}_0''(\\tilde{x}_2) & -\\mathrm{L}_1''(\\tilde{x}_2) & \\cdots & -\\mathrm{L}_{N-1}''(\\tilde{x}_2) \\\\\n",
"\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
"-\\mathrm{L}_0''(\\tilde{x}_{N-2}) & -\\mathrm{L}_1''(\\tilde{x}_{N-2}) & \\cdots & -\\mathrm{L}_{N-1}''(\\tilde{x}_{N-2}) \\\\\n",
"0 & 0 & \\cdots & 1\n",
"\\end{pmatrix}\n",
"\\begin{pmatrix}\n",
"c_0 \\\\\n",
"c_1 \\\\\n",
"c_2 \\\\\n",
"\\vdots \\\\\n",
"c_{N-2} \\\\\n",
"c_{N-1}\n",
"\\end{pmatrix} =\n",
"\\begin{pmatrix}\n",
"\\mathcal{T}_a \\\\\n",
"f(\\tilde{x}_1) \\\\\n",
"f(\\tilde{x}_2) \\\\\n",
"\\vdots \\\\\n",
"f(\\tilde{x}_{N-2}) \\\\\n",
"\\mathcal{T}_b\n",
"\\end{pmatrix} \\quad .\n",
"$$\n",
"\n",
"Let $X = \\{\\tilde{x}_0, \\tilde{x}_1, \\cdots, \\tilde{x}_{N-1}\\}$. The second derivative of the Lagrange polynomials can then be expressed by\n",
"\n",
"$$\n",
"\\mathrm{L}_i''(\\tilde{x}) = \\sum_{\\substack{x' \\in X \\setminus \\{\\tilde{x}_i\\} \\\\ x'' \\in X \\setminus \\{\\tilde{x}_i, x'\\}}} \\frac{\\mathrm{L}_i(\\tilde{x})}{(\\tilde{x} - x')(\\tilde{x} - x'')} \\quad .\n",
"$$\n",
"\n",
"This expression gets easier to evaluate numerically when one only considers $\\tilde{x} \\in X$. In this case, the expression can be rewritten to avoid division by zero.\n",
"\n",
"When $i = j$,\n",
"\n",
"$$\n",
"\\mathrm{L}_i''(\\tilde{x}_i) = \\sum_{\\substack{x' \\in X \\setminus \\{\\tilde{x}_i\\} \\\\ x'' \\in X \\setminus \\{\\tilde{x}_i, x'\\}}} \\frac{2}{(\\tilde{x}_i - x')(\\tilde{x}_i - x'')} \\quad .\n",
"$$\n",
"\n",
"When $i \\neq j$,\n",
"\n",
"$$\n",
"\\mathrm{L}_i''(\\tilde{x}_j) = \\frac{2}{\\tilde{x}_i - \\tilde{x}_j} \\prod_{\\, x' \\in X \\setminus \\{ \\tilde{x}_i, \\tilde{x}_j \\}} \\left(\\frac{\\tilde{x}_j - x'}{\\tilde{x}_i - x'}\\right) \\sum_{x'' \\in X \\setminus \\{ \\tilde{x}_i, \\tilde{x}_j \\}} \\frac{1}{\\tilde{x}_j - x''} \\quad .\n",
"$$\n",
"\n",
"Deriving the expression for the second derivative of the Lagrange polynomials is left as an exercise for the reader (Hint: logarithmic differentiation).\n",
"\n",
"The heat equation is solved by finding the coefficients of the Lagrange polynomials, which is done by solving the system of linear equations that is described above."
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"### Implementing and verifying the solver\n",
"Now it is time to write some code. We begin by implementing functions that evaluate the Lagrange polynomials and their second derivative, as well as functions that solve the system of linear equations and the heat equation."
]
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{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import itertools\n",
"from scipy.integrate import simps\n",
"import scipy.linalg.lapack as lapack\n",
"\n",
"# Set some figure parameters\n",
"newparams = {\n",
" 'figure.figsize': (16, 6), 'axes.grid': True,\n",
" 'lines.linewidth': 1.5, 'font.size': 19, 'lines.markersize' : 10,\n",
" 'mathtext.fontset': 'stix', 'font.family': 'STIXGeneral'}\n",
"plt.rcParams.update(newparams)\n",
"\n",
"def lagrangepoly(xarr, i, x):\n",
" \"\"\" Evaluates the Lagrange polynomial L_i(x).\n",
" \n",
" Parameters:\n",
" xarr: float array. Contains the points that define the particular set of Lagrange polynomials.\n",
" i: int (positive). Selects the Lagrange polynomial corresponding to the point xarr[i].\n",
" x: float. The point at which to evaluate the polynomial.\n",
" \n",
" Returns: The value of the Lagrange polynomial evaluated at x.\n",
" \"\"\"\n",
" xs = np.delete(xarr, i)\n",
" return np.prod(x - xs) / np.prod(xarr[i] - xs)\n",
"\n",
"def lagrangepoly_2nd_deriv(xarr, i, j):\n",
" \"\"\" Computes the second derivative of the Lagrange polynomial, L''_i(x_j).\n",
" This function is limited to evaluating the second derivative at points contained in xarr only.\n",
" \n",
" Parameters:\n",
" xarr: float array. Contains the points that define the particular set of Lagrange polynomials.\n",
" i: int (positive). Selects the Lagrange polynomial corresponding to the point xarr[i].\n",
" j: int (positive). Selects the point xarr[j] at which to evaluate the second derivative.\n",
" \n",
" Returns: The value of the second derivative of the Lagrange polynomial evaluated at xarr[j].\n",
" \"\"\"\n",
" res = 0.\n",
" if i == j:\n",
" # Exclude element x[i] to avoid zero division\n",
" xs = np.delete(xarr, [i])\n",
" # Sum over all combinations of two elements in xs\n",
" for p in itertools.combinations(xs, 2):\n",
" res += 1. / ((xarr[j] - p[0])*(xarr[j] - p[1]))\n",
" else:\n",
" xs = np.delete(xarr, [i, j])\n",
" res = 1. / (xarr[i] - xarr[j]) * np.prod(xarr[j] - xs) / np.prod(xarr[i] - xs) * np.sum(1. / (xarr[j] - xs))\n",
" \n",
" res *= 2\n",
" return res\n",
"\n",
"def compute_coefficients(xarr, farr):\n",
" \"\"\" Computes the coefficients of the Lagrange polynomials defined by the points in xarr,\n",
" essentially solving the heat equation.\n",
" \n",
" Parameters:\n",
" xarr: float array. Contains the points that define the particular set of Lagrange polynomials.\n",
" farr: float array. The values of the heat source term evaluated at the points in xarr.\n",
" \n",
" Returns: A float array containing the coefficients.\n",
" \"\"\"\n",
" N = len(xarr)\n",
" # Initialise matrix A (LHS)\n",
" A = np.zeros((N, N), dtype=np.float)\n",
" for i in range(1, N-1):\n",
" for j in range(0, N):\n",
" A[i, j] = -lagrangepoly_2nd_deriv(xarr, j, i) \n",
" A[0, 0] = 1.\n",
" A[N-1, N-1] = 1.\n",
"\n",
" # Solve the system of linear equations\n",
" carr = np.linalg.solve(A, farr)\n",
" return carr\n",
"\n",
"def solve_heat_equation(xs, N=40, T0=0.0, T1=0.0, x0=0.5, sigma=0.1, chebyshev=True):\n",
" \"\"\" Solves the heat equation by expansion in a basis of Lagrange polynomials.\n",
" \n",
" Parameters:\n",
" xs: float array. An array containing the points at which the solution should be evaluated.\n",
" N: int (positive). The number of collocation points.\n",
" T0: float. Temperature at x=0.\n",
" T1: float. Temperature at x=1.\n",
" x0: float. Mean of the Gaussian heat source.\n",
" sigma: float. Standard deviation of the Gaussian heat source.\n",
" chebyshev: boolean. Use chebyshev nodes as collocation points if true. Use linearly spaced points otherwise.\n",
" \n",
" Returns: A float array of the same dimension as xs containing the temperatures.\n",
" \"\"\"\n",
" # Define collocation points.\n",
" if chebyshev:\n",
" xarr = 0.5 - 0.5*np.cos(np.arange(N)*np.pi/(N-1)) # Chebyshev nodes\n",
" else: \n",
" xarr = np.linspace(0, 1, N) # Equally spaced points\n",
" \n",
" # Initialise farr containing the values of the heat source term (RHS)\n",
" farr = np.exp(-0.5 * (xarr - x0)**2 / sigma**2)\n",
" farr[0] = T0 # boundary conditions\n",
" farr[N-1] = T1 #\n",
" \n",
" carr = compute_coefficients(xarr, farr)\n",
" \n",
" ntemps = len(xs)\n",
" temps = np.zeros(ntemps)\n",
" \n",
" for i in range(0, N):\n",
" for k in range(0, M):\n",
" temps[k] += carr[i]*lagrangepoly(xarr, i, xs[k])\n",
" return temps"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us do a quick test where we solve the heat equation when the temperature is zero at the boundary and the heat source is centered at $\\tilde{x}=0.5$ ."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
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r8+HPCdzy3m8MvbEGQzvUwM9bTe1ERIoyFdEiIlIkHUxK5YU5G/h5y1Eiywbx+eBWRNfIWXne4nA2EQAvD8OjN9bg7iYVeP3bzXzwcwKzVu/n9W4NuaFWGafjiYjIVdKf6UVEpEjJyrZMitvFre/+RvyOY/yzU13mPdH+jwJaxNWUD/bno/ua8fngVnh7ePDAJ8v46/Q1HD+T7nQ0ERG5ClqJFhGRImPL4ZM8M3M9a/YlcUOtMrzerYEeJSRFRnSNMOY90Z5Rv25ndOwOYrf9zot31ePuJhUwxjgdT0RE8kkr0SIi4vLSMrL47/yt3PXhIvYeT+H9vk2YPLCFCmgpcvy8PXnqttrMfbwdlUMDeHL6GgZMXM6+42o8JiJSVKiIFhERl7Zk5zE6fbCQj37dTtfGFfjprzfSrWlFrdxJkVanXElmDo3mpS71WLH7OLe9t4CPF+4kMyvb6WgiInIZupxbRERc0qm0DP49bwtTl+2lUqg/nw5qqWZMUqx4ehgGtq3GbfXL8cKcDbz27Wa+XnuQt3s1pna5Ek7HExGRi9BKtIiIuJy47Ync8f5Cpi3fy8PtqjH/yRtUQEuxVTHEnwkDohh5b1MOnEily8hFjPp1u1alRURclFaiRUTEZZw5m8l/vtvCZ0v2UC0skBmPtqF5lVCnY4kUOmMMXRpXoE2N0rwwZwNvz9/KD5uO8E7vRtQsq1VpERFXopVoERFxCUt3HuPODxbyv6V7GNS2GvMeb68CWtxOWJAvMf2aMfLepuw5doZOHy5i/IKdZGVbp6OJiEgurUSLiIijUtOzeGv+FibF76ZSqQCmDW5Nq+qlnY4l4phzq9Ktqofy3KwNvD5vM/M3Hubt3o2pFhbodDwREbenlWgREXHMyj3H6fThQibG7eaB1lX4/sn2KqBFcpUt4cf4B5rzXt/GbDtyijs/WMDEuF1ka1VaRMRRWokWEZHrLiMrm/d/2sbo2B2UD/bn84dbEV0zzOlYIi7HGEP3phFE1wjjmZnrePmbTfy46Qjv9GlM+WB/p+OJiLglrUSLiMh1teP30/QcHc+oX3fQs1kE8/9ygwpokcsIL+nHJw+24D89GrJ6bxJ3vL+Qb9cdcjqWiIhb0kq0iIhcF9ZaPl+2l9fmbsbX24PR/ZpxZ8PyTscSKTKMMdzTsjKtqpfmyelrGP75Kn7eUpGXu9anhJ+30/FERNyGimgRESl0iafP8szMdfy0+SjtI8P4b+/GhJf0czqWSJF07vFvI3/Zzke/JLBs13He79uEqKrqZi8icj3ocm4RESlUv245yh3vL2BBQiIv3lWPyQNbqoAWuUbenh789dZafPloNB7G0GfsYt75YSsZWdlORxMRKfZURIuISKFIy8jihTkbGDhpOWFBvnw9oi2D2lXDw8M4HU2k2GhepRTznmhPz2YRjPxlO71Gx7Mr8YzTsUREijUV0SIiUuC2HTlF148W8dmSPTzcrhpzhrelTrmSTscSKZaCfL14u3djYvo1Y/exFDp/uJBZq/Y7HUtEpNjSPdEiIlJgrLVMXbaPV+ZuJMjXi8mDWnJjrTJOxxJxC50alqdp5RCemLaGv36xlkUJibzSrQFBvvq6JyJSkPSpKiIiBSI5NYPnZq3n2/WHaB8Zxjt9GlO2hO59Frmeygf7M3Vwa0b+ksCHPyewel8SI+9tSoOKwU5HExEpNnQ5t4iIXLNVe0/Q+cOFzN94mH/cUYfJA1uqgBZxiKeH4clbavH54NakpmfRPSaOTxbtwlrrdDQRkWJBRbSIiFy17GzL6Ngd9B6zGIAvHm3D0A411DxMxAW0rl6a755oz421yvDK3E08PHkFx8+kOx1LRKTIUxEtIiJX5fdTZxkwcRlvfr+FO+qX49vH29OscimnY4nIeUoF+jD+gShe6lKPhQmJ3PnBAhbvOOZ0LBGRIk1FtIiIXLHFO47R6cOFLNt1nDd6NOSj+5oS7O/tdCwRuQBjDAPbVmPWsGgCfbzo9/ESPvolgexsXd4tInI1VESLiEi+ZWdbRv26nX4fL6GEnxdfjWjLvS0rY4wu3xZxdQ0qBvP1Y+24q1EF/vvDNh6ctFyXd4uIXAUV0SIiki/Hz6QzcNJy3p6/lS6NK/D1iHZ69rNIERPk68UH9zTh9e4NWLLzGJ0+WMiK3cedjiUiUqSoiBYRkctasfs4nT5YyOKdx3i9ewPe79tEz54VKaKMMfRrVYVZQ6Px9fag77gljFuwQ927RUTySUW0iIhclLWWcQt20HfcEny9PZg1NJp+raro8m2RYqBBxWC+eawdt9UL59/ztjD405Ukp2Q4HUtExOWpiBYRkQtKTslg8Kcr+fe8LdxWL5xvHmtHg4rBTscSkQJU0s+bmH7NeKlLPX7bdpTOIxeydl+S07FERFyaimgREfmTDQeS6TxyIb9tO8pLXeoR068ZJf3UfVukODrXvfvLR6OxFnqNied/S/bo8m4RkYtQES0iIv/HF8v30WN0PFnZlumPtGFg22q6fFvEDTSpFMK3j7ejbc0wnp+zgae+XEtqepbTsUREXI6KaBERASAtI4tnZq7j6ZnraFk1lLmPtaNZ5VJOxxKR6ygkwIdPBrTgL7fUYvbqA3SPiWN34hmnY4mIuBQV0SIiwr7jKfQes5hpy/cxvGMNJg9qSekgX6djiYgDPDwMT9wSycQHW3D4ZBpdPlrEj5uOOB1LRMRlqIgWEXFzsVuP0uWjRew+dobxD0Tx99vr4Omhy7dF3F2H2mX5ZkQ7qpYOZPCnK3jr+y1kZes+aRERFdEiIm4qO9vy/k/bGDhpOeVK+vHNiHbcWi/c6Vgi4kIqhQbw5aNtuLdlJWJid/DAJ0s5dvqs07FERBylIlpExA0lp2Tw0OTlvP9TAt2bVmT2sLZUDQt0OpaIuCA/b0/e6NGIt3o1YvnuE9w1chFr9BgsEXFjKqJFRNzMlsMn6TpqEYu2J/Jqtwa807sx/j6eTscSERfXJ6oSs4ZG4+lh6DNmMdOX73U6koiII7ycDpCXMaYu8CKwA6gGHAaes9Ze9NohY4w/8DwQAqQBkcCz1tqNl5jTFFhmrdWDT0XEbcxdd5C/f7mOEn5eTBvShuZV1H1bRPKvQcVgvhnRjsemruYfM9ezbn8yL3Wpj4+X1mVExH24VBFtjKkNxAMPWmu/yh37GpgK9LjIHA/gSyDdWtsjd6wnsMAY09xau/sCc7yBibjY8YuIFJbMrGzenr+VsQt2ElWlFDH9mlG2pJ/TsUSkCCoV6MPkQS15e/5Wxvy2gy2HTxHTrxnh+kwRETfhan82HAkkniugc/0X6G6Muesic+4BOgPvnBuw1s4EknLnXsjTgJ7VICJu4fiZdAZMXMbYBTt5oE0VPh/cWgW0iFwTTw/DM3fWYdR9zdh86CR3jVzEit3HnY4lInJduEwRbYypBNwKLMizaTGQDgy+yNRBwFlgaZ7xBUA3Y0yZPO/TACiZ+3tFRIq1DQeS6TJyEct3n+DtXo145e4GuuxSRApM50blmT2sLYE+ntw7fgmfLdmDtXoMlogUb670Tap17uv28wettRnAPqB93gnGGAO0AvZYazPzbN4BeALR5+3vCbwAvFxwsUVEXNOsVfvpOToeay0zHm1D76hKTkcSkWKodrkSfDWiHe1qhvHCnA08PWMdaRlZTscSESk0rlRER+S+HrvAtkSglDEmbwecECDoEnMAqp839hQwzlqbci1BRURcWWZWNq98s4m/frGWppVD+PqxdjSKCHE6logUY8H+3kwY0ILHb47ky5X76TtuCYeT05yOJSJSKFypsZZ/7mvGBbal574GAieuYg7GmDpAJWvtW/kNZIwZAgwBCA8PJzY2Nr9Tr7vTp0+7dD5xHzoXnXU63RKzNo1Nx7K5tYoXfWumsWGF+929ovNQXIE7nofNvOGxpr6MX5fE7e/+wmNNfKlZSo/Qc5o7novieorTeehKRfS5leMLPXLqXM4zecbPrUBfck5uB+9XgYevJJC1dhwwDiAqKsp26NDhSqZfV7GxsbhyPnEfOheds+XwSQZ/uoIjyfB2r0Zuffm2zkNxBe56HnYAunQ4xeBPV/DWijRe7Vafvi0qOx3LrbnruSiupTidh650Ofeh3NcLPbS0NJBsrT1/FZrcZ0cfv8QcgF3AE8AXwFljjJ8xxo/cIvu8n0VEiqzv1h+iR0w8ZzOymf5Ia7cuoEXEebXCS/D18Ha0qh7KP2au58WvNpCRle10LBGRAuFKK9HLAQvUOH/QGOMFVAa+vsi8pUBHY4yntfb8LhY1yOnaHQs8Cdx4kfmp597q6mKLiDgnO9vy/k/b+PCX7TStHMKY+5vrWa0i4hKCA7yZNLAlb32/hbELdrI193nSpYN8nY4mInJNXGYl2lp7GPiFP3fhbgn4AVMAjDFBeRqMfZ67PSrPvBuAedbaJGAY0CbPfxNy9zv3s4hIkXIqLYMhn63kw1+20ycqgmlDWquAFhGX4ulheLZTXd7v24Q1+5Lo+lEcGw8mOx1LROSauEwRnetvQBVjTKc8YwuAubk/rwS2G2MCc3+eCsSR03kbAGPM3eR0+34JwFq7yVq75Pz/gP252879LCJSZOxKPEP3mHh+3XqUl7vW582ejfD1UvMeEXFN3ZpWZMaj0WRbS8/R8Xyz9qDTkURErporXc6NtXaNMaYD8Hzua1ngANDfWnvuRpoD5HTjzsydk2WMuRN4wxgzkZwGZbWBG6y166/zIYiIFLqFCb8zfMoqPD0M/3uoFW1qlL78JBERhzWMCObrEe0YNmUlj01dzdbDp/jrrbXw8NAddSJStLhUEQ1grV0GdL3E9psuMHYKGHGF7/Mv4F9XGE9ExDHWWibF7+a1bzdTs0wQHw+IolJogNOxRETyrUwJX6Y83JoXv9rAR79uZ+uRU7zXtwlBvi73lVRE5KJc7XJuERG5gLOZWTwzcz0vf7OJm+qUZeawaBXQIlIk+Xh58EaPhvyrSz1+2XKUnjHx7Due4nQsEZF8UxEtIuLiEk+fpd/4pUxfsY/HbqrJ2Puba9VGRIo0YwwPtq3G5IEtOXwyja4fLWLxjmNOxxIRyRcV0SIiLmzjwWS6jlzEhoPJjLy3KU/dVlv3D4pIsdEuMow5w9tSOsiX/hOW8tmSPU5HEhG5LBXRIiIuat76Q/QavRgLfPlINF0aV3A6kohIgasWFsisYdG0jwzjhTkbeH7OejKysi8/UUTEISqiRURcTHa25f2ftjFsyirqli/BVyPa0jAi2OlYIiKFpqSfNx8PaMEjN1bnf0v20n/CUk6cSXc6lojIBamIFhFxIanpWTw2dTXv/5RAz2YRTB3SmrIl/JyOJSJS6Dw9DM/eWZd3+zRm1d4k7h4VR8KRU07HEhH5ExXRIiIu4nByGn3GLmbehkM8e2cd/tu7Eb5enk7HEhG5rno0i2Dq4NakpGfSIyaeX7cedTqSiMj/oSJaRMQFrN2XRNePFrHz99OM7x/FIzfWwBg1EBMR99S8Sim+GtGOiNAAHpq0nI8X7sRa63QsERFARbSIiOO+XnuQPmMX4+Plwcxh0dxSL9zpSCIijqsY4s+MR9twa71wXvt2M8/MXE96phqOiYjzVESLiDgkO9vyzg9beXzqahpHhPDV8LbUKVfS6VgiIi4j0NeL0f2aM6JjTaav2Mf9E5ZyXA3HRMRhKqJFRByQkp7J8M9XMfKX7fSJiuB/D7eidJCv07FERFyOh4fhb7fX5oN7mrBmXxJ3j1rENjUcExEHqYgWEbnOzjUQ+37jYZ7vXJc3ezbCx0sfxyIil3J3k4pMH9KatIzsnIZjW9RwTEScoW9tIiLX0br9OQ3EdiemMGFAFA+3r64GYiIi+dS0cim+Gt6WyqEBPDR5ORMW7VLDMRG57lREi4hcJ/PWH6LP2MV4e3owY2gbbqqjBmIiIleqQog/M4a24Za64bw6dxP/nLOBjCw1HBOR60dFtIhIIbPWMvLnBIZNWUW98iX5aoQaiImIXIsAHy/G3N+coR1q8PnSvTw4cRnJKRlOxxIRN6EiWkSkEKVlZPGX6Wt458dtdGtSgc8HtyZMDcRERK6Zh4fhH3fU4e1ejVi26zjdY+LYlXjG6Vgi4gZURIsFeOwDAAAgAElEQVSIFJLfT53lvvFLmLPmIH+7rRbv9W2Cn7en07FERIqV3lGVmPJwa06kpNNtVBzxOxKdjiQixZyKaBGRQrDl8Em6jYpj06GTxPRrxoibItVATESkkLSsFspXw9tRtoQvD0xYxtRle52OJCLFmIpoEZEC9uuWo/SMiSczO5svHmlDp4blnY4kIlLsVS4dwMxh0bStGcazs9bz2txNZGWrc7eIFDwV0SIiBcRay8S4XTw0eTlVwwKZM7wtjSJCnI4lIuI2Svp5M2FAFA9GV+XjRbt45LOVnDmb6XQsESlmVESLiBSAzKxsXvxqIy9/s4mb64bzxSNtKB/s73QsERG34+Xpwb+61uflrvX5ZcsReo9ZzKHkVKdjiUgxoiJaROQanUzLYOCk5Xy2ZA+P3FCdsfc3J9DXy+lYIiJubUB0VT55sAV7j6dw90dxrNuf5HQkESkmVESLiFyDvcdS6BkTz+Idx3izZ0Oe7VQXDw81EBMRcQUdapdl5tBovD096DN2Md+tP+R0JBEpBlREi4hcpRW7j9MtJo6jp87y6UMt6duistORREQkj9rlSjBneFvqli/J0CmrGPXrdqxVwzERuXoqokVErsKc1Qe4b/xSSvp5MXtYNNE1wpyOJCIiF1GmhC9TB7ema+MKvD1/K3+fsY70zGynY4lIEaWb9kREroC1lvd/SuCDnxNoVS2UMfc3p1Sgj9OxRETkMvy8PfngniZULxPI+z8lsO94ij7DReSqaCVaRCSf0jKyeGLaGj74OYFezSP47KFW+vIlIlKEGGN48pZafHBPE1bvTaLH6Hh2JZ5xOpaIFDEqokVE8uHY6bPc//FSvl57kKfvqM3bvRrh46WPUBGRoujuJhX5fHArklMz6B4Tx5Kdx5yOJCJFiL4Biohcxvajp+keE8/6A8mMuq8ZwzrUxBh14BYRKcqiqoYye1g0pQN96D9hKTNX7nc6kogUESqiRUQuIW57It1j4khJz2TakNZ0blTe6UgiIlJAqpQOZNbQtrSoGspTX67lv/O3kp2tzt0icmkqokVELmL68r0M+GQZ5YP9mD2sLU0rl3I6koiIFLDgAG8mD2pJ36hKfPTrdh6ftpq0jCynY4mIC1N3bhGRPLKzLW/N38qY33ZwQ60yfHRfU0r6eTsdS0RECom3pwf/6dmQ6mUCeeO7LRxISmVc/yjKlPB1OpqIuCCtRIuInCc1PYthU1Yx5rcd3N+6Mp8MiFIBLSLiBowxPHJjDcbc34zNh07SPSaOhCOnnI4lIi5IRbSISK6jp9K4Z9xi5m86zPOd6/Lq3Q3w8tTHpIiIO7mjQXmmD2nD2cxsesTEsygh0elIIuJi9O1QRATYevgU3UfFs+3Iacbe35yH21dXB24RETfVuFIIc4a3pUKIPw9OXMa0ZXudjiQiLkRFtIi4vQXbfqfX6HgysrL54pE23Fa/nNORRETEYRVD/JkxtA3RNcN4ZtZ63vhuszp3iwigIlpE3NyUpXsYOGk5FUv5M2d4WxpGBDsdSUREXEQJP28+GRBFv1aVGfvbToZ/vorUdHXuFnF3KqJFxC1lZ1te/3YT/5y9gfaRYcwYGk2FEH+nY4mIiIvx8vTgtW4NeL5zXb7feJh7xi/h6Kk0p2OJiINURIuI20lNz2LolJWMX7iLB9pU4eMHogjy1RP/RETkwowxPNy+OmPub8623B4aWw+rc7eIu1IRLSJu5eipNPqOW8wPm47w4l31eLlrfXXgFhGRfLm9fjm+eKQN6VnZ9Bodz8KE352OJCIO0DdHEXEb5zpwJxw5zbj+UQxqV00duEVE5Io0jAhmzvC2VCzlz8CJy9W5W8QNqYgWEbewMOH/duC+tV6405FERKSIqhjiz5eP/v/O3W9+v0Wdu0XciIpoESn2pi7by4MT1YFbREQKTgk/byYMiOK+VpUZHbuDx6auJi1DnbtF3IHLddIxxtQFXgR2ANWAw8Bz1tqzl5jjDzwPhABpQCTwrLV2Y5797gGeBWoCe4HXrLVTCuM4RMR52dmWt+ZvZcxvO7ixVhk+uq8pJfy8nY4lIiLFhLenB693a0DV0gG88d0WDianMv6BKMKCfJ2OJiKFyKVWoo0xtYF4YJq19nlrbT9yCuKpl5jjAXwJ1LXWDrfWPgVMBhYYY6qet18/oAvwGHAXOcX5Z8aYVoV0OCLioLSMLEZMXcWY33bQr1VlJgyIUgEtIiIFzhjDkBtqMLpfMzYfOkn3mDi2H1XnbpHizKWKaGAkkGit/eq8sf8C3Y0xd11kzj1AZ+CdcwPW2plAUu5cjDHeQHVrbT9r7QJr7a/AA4AB2hf8YYiIkxJPn+WecUv4bsNhnu9cl9e6NVAHbhERKVR3NCjPtCFtSE3PontMPPHbE52OJCKFxGW+VRpjKgG3AgvybFoMpAODLzJ1EHAWWJpnfAHQzRhTBrDAm3m2n8x9XX+1mUXE9Ww/eopuo+LYcvgko/s15+H21dWBW0RErosmlUKYPawt5Ur68cAny5ixcr/TkUSkELhMEQ20zn3dfv6gtTYD2McFVoxNzjfjVsAea21mns07AE8g2lqbaa1Nz7P9AeBDa+38gggvIs6L35FIj5h40jKymT6kDXc0KOd0JBERcTOVQgOYOSya1tVL87cv1/LuD1uxVp27RYoTV2osFpH7euwC2xKBGsaYUtbaE+eNhwBBl5gDUP38QWNMEPAE8DdymoxdlDFmCDAEIDw8nNjY2MscgnNOnz7t0vnEfTh1Li46kMHEDemEBxr+0sybEzvWELvjuscQF6HPRHEFOg/d24PVLSbViw9/2c6yzbt4qKEv3h7OXBmlc1FcQXE6D12piPbPfc24wLZzq8iBwPlFdH7nAH908X4CqEPOKvVoY0yotfbfFwpkrR0HjAOIioqyHTp0uPxROCQ2NhZXzifu43qfi9Za3vspgY/XJ9CuZhij+jUj2F8NxNydPhPFFeg8lJs7WmJid/D2/K1k+/oxtn9zSgX6XPccOhfFFRSn89CVLuc+t3J8oW+/54r9M3nGz61A52uOtTbVWvu6tbY/OY+52gr84+riiojTzmZm8Zfpa/jw5wT6REUwcWALFdAiIuIyjDEM71iTD+9typp9SfQYHc/uxLxfZ0WkqHGlIvpQ7mupC2wrDSTnuZSb3GdHH7/EHIBdF3oza+1R4F2gZG7zMREpQpJS0uk/YRlz1hzk77fX5s2ejfBWB24REXFBXRtXYMrgViSlpNM9Jo4Vu487HUlEroErfeNcTk4X7RrnDxpjvIDKwMUagC0FqhpjPPOM1yCna3fsJd5zX+4+F7qnWkRc1O7EM/SIiWfN3iQ+uKcJwzvWVAduERFxaS2qhjJrWFtCAny47+OlfLP2oNORROQquUwRba09DPzCn7twtwT8gCmQ0xjMGHP+yvPnuduj8sy7AZhnrU26xNs2BmZYa7OvJbuIXD8r9xynx+h4TqSkM2VwK+5uUtHpSCIiIvlSLSyQWUOjaRwRzGNTVzPq1+3q3C1SBLlMEZ3rb0AVY0ynPGMLgLm5P68EthtjzjUMmwrEAU+dm2CMuZucbt8v5f5c3hjzqzFmRO7KNsaYekA34OlCPB4RKUDfrjvEveOXUtLPi1nD2tKiaqjTkURERK5IqUAfPnuoFV0bV+Dt+Vt5ZuZ6MrK0niNSlFxTd25jjA85q7mVyWnudQzYaK29qutTrLVrjDEdgOdzX8sCB4D+560WHyCnG3dm7pwsY8ydwBvGmInkNCirDdxgrV2fOyctN99/geHGmJ+BvcDt1trkq8kqItePtZYxv+3kze+3EFWlFOMeiCLUge6mIiIiBcHP25P3+zahcmgAH/26nYPJqYzq14ySfmqOKVIUXFURbYxpDPwduBsIAM7djGhzt68HRgITr/RSaWvtMqDrJbbfdIGxU8CIS8w5AbS7khwi4hoysrJ58asNTF22jy6NK/B2r0b4eedtgSAiIlK0eHgY/nZ7bSqHBvDc7PX0Hr2YTwa2oGKI/+Uni4ijruhybmOMhzHmFWAOkAo8C/QCbgduBroADwHfAY8AS40xlQs0sYi4jVNpGQyatJypy/YxvGMNPujbRAW0iIgUK31aVGLSwJYcTEql26g41u/XRZIiri7fK9G5l25PAWYDNfKzwmyMaQNMMMY8Yq3defUxRcTdHExKZdCk5SQcPc2bPRvSt4X+HiciIsVTu8gwZgyNZtCk5fQZu5iR9zbllnrhTscSkYu4kpXofwD/sNZ+nt9LtK21i4EewFPGGF2bIiL5suFAMt1GxXHgRCqTBrZQAS0iIsVe7XIlmD0smpplgxjy2Qomxe1yOpKIXES+iujcTtifn7+abIzpZYypdLm5ufcr/52cbtkiIpf0y5Yj9Bm7GC8Pw4yh0bSPLON0JBERkeuibEk/pj/SmpvqhPOvbzbx8jcbycrWI7BEXE1+V6KfBvKuPo8CovPuaIypY4zpc/6YtTbFWptwdRFFxF18tng3D09eQfUygcwZ3pba5Uo4HUlEROS6CvDxYmz/5jwYXZWJcbt59H8rSUnPdDqWiJwnv0X0J8AiY8z5HX3+DgTl3dFauwU4bYx5rwDyiYgbyM62vDZ3Ey98tZGb6pRl+pA2lC3p53QsERERR3h6GP7VtT4vdanHT5uPcO+4Jfx+6qzTsUQkV36L6BCgPBB4bsBa+ymw3RjznDGmrDGmnjHm4dxnNU8AHizwtCJS7KSmZzFsyio+XrSLB6OrMrZ/FIG+1/QIexERkWJhYNtqjOsfxbYjp+keE0fCkVNORxIR8l9EPwt8b609eW7AGNMO6EbOI64OAevJucS7ATCNnMdeiYhcVOLps9w7fgnzNx3mxbvq8a+u9fH0MJefKCIi4iZurRfO9Edak5aRTY/R8cTvSHQ6kojby28R3Q54N8/Yt8B9wE7gGaAvMAnobq39i7V2WUGFFJHiZ/vRnL+qbzl8kjH3N2dQu2pORxIREXFJjSJCmDM8mnIl/RjwyTJmrtzvdCQRt5bfInouEJVn7AuggrW2l7X2bWvtDOAx4D5jTO+CDCkixcuSncfoERNHanoW04a04fb65ZyOJCIi4tIiSgUwY2g0LaqG8tSXa3nvx21Yq87dIk7IVxFtrX0UWG+MOb9V7hfW2qw8+6Vba98CPIwx/yq4mCJSXMxevZ/+E5ZStqQfs4e1pUmlEKcjiYiIFAnB/t5MGtiSXs0j+ODnBJ76Yi3pmXkfoCMihS2/K9FYa7/NfebzuZ9/vMTu5YAnryWYiBQv1lo+/DmBv0xfS1SVUGY+Gk2l0ACnY4mIiBQpPl4evN2rEX+9tRazVh9gwCfLSE7JcDqWiFvJdxF9hX4GhhXS7xaRIiY9M5u/z1jHuz9uo0ezikwe1JLgAG+nY4mIiBRJxhgevzmS9/o2ZsWe4/QcE8++4ylOxxJxG/kqoo0xJYwxffP7S621G6y1n583v6kxpsXVBBSRoi05NYOBk5YxY+V+nrwlknd6N8bHq7D+ficiIuI+ujeN4LOHWvH7qbN0j4lj7b4kpyOJuIX83hN9Cggzxjx2pW9gjLkV+Iu1dvmVzhWRom3/iRR6j4ln2a7jvNO7MU/eUgtj9AgrERGRgtK6emlmDo3G38eTvuMW88PGw05HEin2ruSe6FFAZWPMPGNMq8vtb4xpYoyZBDwNPHr1EUWkKFq3P4nuMfEcSk5j8sCW9Gwe4XQkERGRYqlm2SBmD2tL7XIleeR/K/lk0S6nI4kUa1752ckY80/gHWvt340xjwA/GmPSgA3AQeBM7q6BQHmgAVASeAt42FqbWeDJRcRl/bTpCI9NXU1ooA+fP9yKyPASl58kIiIiVy0syJdpg1vz5PTVvDJ3E3uPp/DCXfXw9NAVYCIFLV9FNLAcmAZ0s9aONcbMJKdx2N1Au/N+TxY5hfUYYKy19mAB5xURF/fTngw+n7+CBhWD+XhAFGVL+DkdSURExC34+3gS0685b8zbzMeLdrH/RCof3tvE6VgixU5+i+ifgdnGmC7W2m+stYnAK8ArxhgvoDRggGPWWvXYF3FDWdmWf8/bzP82p3NL3XA+vLcJAT75/YgRERGRguDpYXj+rnpUCg3g5W82cs+4JQyK1LOkRQpSfhuLZQGHgQEX2JZprT1irT2sAlrEPaWmZzFsykomLNrFrVW8GNu/uQpoERERBw2Irsq4/lEkHDnNq4vTSDhyyulIIsXGlT5nJrpQUohIkfX7qbPcM34JP2w6wot31aNfXV/dfyUiIuICbqkXzhePtCHTQo/R8cRvT3Q6kkixcCVFdDAQbowpVVhhRKRo2X70ND1Gx7H18EnG3t+cQe2qOR1JREREztMwIpgXWvtRPtiPAROXMXPlfqcjiRR5V7oSDXDIGPOLMeZpY0zDAk8kIkXCkp3H6BETR2p6FtOHtOG2+uWcjiQiIiIXEObvwZePRtOiaihPfbmW93/ahrXW6VgiRdaVFNEnAUtOM7IOwBvAGmPMfmPMx8aYXsaYkoWQUURczJzVB+g/YSllS/oxe1hbGlcKcTqSiIiIXEKwvzeTBrakV/MI3v8pgae+XEt6phqOiVyNKymidwGDyLms+w7gfWArUCF3fDpw1BjzlTFG906LFEPWWkb+nMCT09fQvEopZj4aTaXQAKdjiYiISD74eHnwdq9G/PXWWsxadYABnywjOVV9gUWu1JUU0YuAQGvtGWvtD9bap6y19YDqwHBgHpAJdAEWGGPuKfi4IuKUjKxs/jFzHe/8uI3uTSsyeVBLggO8nY4lIiIiV8AYw+M3R/Jun8as2HOcXqPj2Xc8xelYIkXKlRTRk4FueQettbuttaOttV2AUKAzMBd4tWAiiojTTqZlMHDicr5Ysf+Pf3h9vTydjiUiIiJXqUezCCYPasnhk2l0j4ln3f4kpyOJFBn5LqKttTuBWGNM70vsk26t/c5a2w1YUhABRcRZB5JS6T16MUt2HvvjEjBj9AgrERGRoi66Rhizhkbj6+VB37FL+GnTEacjiRQJV9Sd21r7BlDLGHN7Pvbtf9WpRMQlbDiQTPdRcRxMSmXyoJb0jqrkdCQREREpQJHhJZg9PJrI8CCGfLaCyfG7nY4k4vKu+BFX1trXybk/WkSKsV+2HKHP2MV4e3owY2g0bWuGOR1JRERECkHZEn5MG9Kam+uG89LXG3l17iaysvUILJGLuZrnRGOtPVPQQUTEdXy2ZA8PT15B9TKBzB4WTe1yJZyOJCIiIoUowMeLMfc358HoqkxYtIthU1aSmp7ldCwRl3RVRbSIFE/Z2ZY35m3mhTkb6FC7LNOHtKFsST+nY4mIiMh14Olh+FfX+rx4Vz1+2HSEe8cvIfH0WadjibgcFdEiAkBaRhYjpq5i7IKd9G9dhXH9mxPo6+V0LBEREbnOBrWrxpj7m7Pl8Em6x8Sx4/fTTkcScSkqokWE42fS6ffxUuatP8w/O9Xllbvr4+WpjwcRERF3dXv9ckwb0obU9Cx6xMSzdOcxpyOJuAx9SxZxc7sSz9AjJo4NB5KJ6deMwTdU1yOsREREhCaVQpg1tC2lg3zoP2EZX6054HQkEZegIlrEja3YfZweMXGcTMvk88Gt6dSwvNORRERExIVULh3ArKHRNK0cwhPT1jDq1+1Yq87d4t5URIu4qbnrDnLfx0sJCfBh9rBomlcp5XQkERERcUEhAT58+lBLujWpwNvzt/LsrPVkZGU7HUvEMeoaJOJmrLWMXbCT/3y3hagqpRj/QBSlAn2cjiUiIiIuzNfLk/f6NqFSaAAjf9nOgaRUYvo1o4Sft9PRRK47rUSLuJHMrGyen7OB/3y3hbsaled/D7dSAS0iIiL5Yozhqdtq82bPhsTvOEbvMYs5lJzqdCyR605FtIibOH02k4c/XcGUpXt59MYafHhPU/y8PZ2OJSIiIkVM3xaVmfhgC/afSKXbqDg2Hkx2OpLIdaUiWsQNHDmZRt+xi1mYkMi/uzfkmTvr4OGhDtwiIiJydW6oVYYZQ9vgYQx9xiwmdutRpyOJXDcqokWKuS2HT9JtVBy7E8/w8YAo7mtV2elIIiIiUgzUKVeS2cPaUqV0IA9NXsHnS/c6HUnkunC5ItoYU9cYM9UY85oxZoox5h1jjO9l5vgbY143xozK3f9rY0z9PPsYY8zTxpgdxphUY8xGY8xwowfiSjG2KCGR3qMXk20tXzzaho61yzodSURERIqRcsF+fPFoG9pHhvHc7PW8+f0WsrP1CCwp3lyqO7cxpjYQDzxorf0qd+xrYCrQ4yJzPIAvgXRrbY/csZ7AAmNMc2vt7txdnwduAz4FygL3Ax8BJYD/FNYxiTjlixX7eG7WemqWDeKTB1tQIcTf6UgiIiJSDAX5evHxA1G8+PVGRsfuYP+JVN7u1Ui9V6TYcrWV6JFA4rkCOtd/ge7GmLsuMuceoDPwzrkBa+1MICl3LsaYMkBj4AZr7cvW2uFAK+As8PcCPwoRB1lrefeHrTw9Yx1tapTmi0fbqIAWERGRQuXl6cHr3RrwzJ11+GbtQfpPWMqJM+lOxxIpFC5TRBtjKgG3AgvybFoMpAODLzJ1EDnF8NI84wuAbrkFdDvgWWvtH9eWWGu3AL8BocaYEtd+BCLOS8/M5qkv1vLhL9vp3TyCTx5sQUk9v1FERESuA2MMj95Yg5H3NmXtvmR6jo5nz7EzTscSKXAuU0QDrXNft58/aK3NAPYB7fNOyL2fuRWwx1qbmWfzDsATiLbWzrbWJlzgPU8Ch621p641vIjTklMyeOCTpcxafYC/3lqLt3o1wtvTlf4vLiIiIu6gS+MKTBnciuMp6XSPiWfV3hNORxIpUK70DTsi9/XYBbYlAqWMMaXyjIcAQZeYA1D9Eu9ZF5hwJSFFXNG+4yn0HBPPyj0neK9vYx6/ORL1zBMRERGntKgayqyh0QT5enHvuCV8v+GQ05FECowrNRY7d9NmxgW2nbuhIhA4/09Z+Z3zJ8aYZuQU4BdtKmaMGQIMAQgPDyc2NvZiuzru9OnTLp1PCs+u5CzeW3mWzGzLX5v5USp5O7Gx2y8/sZDoXBRXoPNQXIHOQ3EVTp6Lf2sCH6yEof9bxT11fLitipf+0O+mitNnoisV0edWji90A+e5nHlvqji3An0lc8519P430N1ae/pigay144BxAFFRUbZDhw4X29VxsbGxuHI+KRw/bjrCWz+vpnSQH5MGtqBmWedv79e5KK5A56G4Ap2H4iqcPhdv65jFX6avYeqGw/iGVuCFu+rh6aFC2t04fR4WJFe6nPvcNR55L9kGKA0kW2v/zw0V1tqzwPFLzAHYdYFtbwDvWWtXX2VWEcdNjt/NI5+tIDI8iNnD2rpEAS0iIiKSl5+3J6Pua8bg9tWYFL+bRz5bSUp63nZGIkWHKxXRywEL1Dh/0BjjBVQG5l9k3lKgqjEm74PoapDTtTs2z+97Alhurb3Y7xNxaVnZllfnbuKlrzdyc91wpg1pTZkSvk7HEhEREbkoDw/DPzvX45W76/PLliPcM24JR0+lOR1L5Kq4TBFtrT0M/MKfu3C3/H/t3Xt4V9Wd7/H3SriFIMHIpchNlEJFUTDILTNUHWtbrSJQFKqIoqKCHtuj7Yw9antqa89M9WiPhyB4wytSFC+jzGgdRaYERKkooIKAF0QEQUFQIJCs80d+eDIpYAgJeyd5v56H59fs/VvbT55+c/lm7b0W0Ax4GCCE0KLSAmOPZM73rTRuMDArxrhp94EQwkWUz2g/VvGNIYQzgw9nqA7YVlLK+IcXcs9f3uPCQUdw5/kFNG+SpqcyJEmS9u6CgUcwZXRf3l23laETi3l3nZvkqO5JTROdcS3QJYRweqVjc4BnMh8vBFaEEHYvGDYNmAtcs3tACGEI5at9/6rCsR8DVwMdQwjXV/g3EfhhxT2kpTTasHUHo+6az/NvrePGH/Xk12cd4/NEkiSpzjm1ZzumXzaAHbvKGD6pmOKVG755kJQiqZrCijEuCiGcBFyfeW0LrAFGxxjLMm9bQ/lq3LsyY0pDCD8Efh9CuI/yBcp6AINjjIsBMtd6GGgCHL+H//Sg2vqcpJqw8tOtXHTfq6zfsp1J5xXwg2O/lXQkSZKkajuuYyuenDCIi+57lTH3LuBffnwcQ/t0/OaBUgqkqokGiDEuAM7ax/lT9nBsC3DlPsbMBnxoVHXSgvc+49IHXqNRVmDapQPo03lP6+hJkiTVLR0Pbc5jVwzi8gcX8rPpb7D6s21cdUo3t8BS6qXtdm5JFTy1aA3n3/0Kh7VowhPjC22gJUlSvZKX05j7x/ZjWJ8O/O8/L+cXj73JztKybx4oJSh1M9GSIMZI0eyV/OG5ZfTrms+U0QW0at4k6ViSJEk1rkmjLG4953g65Tfnj//xLms3b6fo/BNo2axx0tGkPXImWkqZnaVl/PKJxfzhuWUM6X04D17czwZakiTVayEEfva97vzhx8cxf9VGRkyax5pN25KOJe2RTbSUIlt37OLi+19j2oLVXHlyN247pzdNG1XeAl2SJKl+GtG3E/eP7cfHm7YxdOJclqzZnHQk6W/YREspsXbzNkbcOY+5Kzbwv4b14trv9yDLLawkSVIDU9itNY9dMYhGWYFzJs/jpXfWJx1J+i9soqUUeHvtFwydWMzqz77i3gtPZGS/zklHkiRJSkyPbx3CkxMKObJNLhff/yoPzf8g6UjS12yipYS9vPxTRtw5D4A/XTaQ73Zvk3AiSZKk5LVt2Yzp4wZyUo+2XP/kEn7/b29TVhaTjiXZREtJmv7qh4yd+iqd8pvz5IRCeh7eMulIkiRJqZHbtBFTRhdw/oDOTH55FVc9+jrbd5YmHUsNnFtcSQmIMXLr88v5vy+tYHD3Nkz8SR8OcRsHSZKkv9EoO4ubhhxL5/zm3DzrHdZt3s6UC/qSn+vuJUqGM9HSQbZjVyk/nW8Cw8oAACAASURBVL6I//vSCkb168Q9Y/raQEuSJO1DCIFxg49i4k9O4M01mxk+qZj3N3yZdCw1UDbR0kG06asSRt+zgKcWfcwvftCDm4f2onG2X4aSJElVccZx7Zl2aX82fVXCsEnFLPzg86QjqQHyt3fpIFn92VcMn1TMog838ceRvRl/UjdCcAsrSZKk/VHQJZ+Z4wtp2awRP7lrPrMWr006khoYm2jpIFi0ehNDi+ayYWsJD17cjyG9OyQdSZIkqc7q2jqXmeMLObZDHhMe+St3zVlFjK7crYPDJlqqZc8t/YSRU+aR0ySbmeMH0f/Iw5KOJEmSVOfl5zbh4Uv6c/qx7fndrLe58aml7CotSzqWGgBX55Zq0b1/eY+bnn2L4zu24u4xfWndomnSkSRJkuqNZo2zuWNUHzoemsPkOav4eNM2/s+oPuQ2tc1R7XEmWqoFpWWRXz+9lN888xan9WzHtEsH2EBLkiTVgqyswHWnH81NZx/LS8vWc+6Ueaz/YnvSsVSP2URLNWxbSSlXPLSQqcXvc/HfdaXovAJymmQnHUuSJKleGz2gC3eP6cuqT79kaFExy9dtSTqS6imbaKkGfbplByOnzOPPb6/j12f25IYf9SQ7yxW4JUmSDoZTvtOOP102kJLSMoZPKqZ4xYakI6kesomWasiK9VsZNmkuy9ZtYfL5BVxY2DXpSJIkSQ3OsR3yeHJCIe3zmnHBvQt4bOFHSUdSPWMTLdWA+as2MqxoLttKSpk+biCnHfOtpCNJkiQ1WB1a5fDYFYPof2Q+1854g9tfWO4WWKoxNtHSAXpq0RouuGcBbQ5pyhPjCzm+U6ukI0mSJDV4LZs15r4L+zH8hI7c/sK7XDvjTUp2uQWWDpxrv0vVFGNk4ksruOX55fTvms+U0X3Ja9446ViSJEnKaNIoi1tGHEfn/Obc9sJy1m7exqTzC8jL8Xc2VZ8z0VI17Cwt47qZi7nl+eWc3ftwHri4nw20JElSCoUQuPrUb3PriONZ8N5njLizmDWbtiUdS3WYTbS0n7Zs38nF97/Go6+u5qpTunHbub1p2sgtrCRJktJseEFHHhjbj7Wbt3P2xLksWbM56Uiqo2yipf2wdvM2Rtw5j7krNvDPw3txzWk9CMEtrCRJkuqCQd1a8/gVg2iSncU5k+fx4jvrko6kOsgmWqqitz7+gqETi/no823cd+GJnHti56QjSZIkaT91b3cIT4wfxJFtcrnk/td4cP4HSUdSHWMTLVXBy8s/ZcSdxQDMuHwgg7u3STiRJEmSqqtty2ZMHzeQk3u05YYnl/D7WW9TVuYWWKoam2jpGzy64EPGTn2Vzofl8uSEQo5u3zLpSJIkSTpAuU0bMXl0AaMHdGHynFVcNe11tu8sTTqW6gC3uJL2IsbILc8vY+JLK/lu9zZMPO8EWjT1S0aSJKm+aJSdxW+GHEPn/Ob8btbbfPLFdu66oC/5uU2SjqYUcyZa2oMdu0r56fRFTHxpJaP6deLuMX1toCVJkuqhEAKXDj6SovNOYPGazQyfVMz7G75MOpZSzCZaqmTTVyWMvmcBTy36mF/8oAc3D+1F42y/VCRJkuqz03u1Z9ql/dn0VQlDi+ay8IPPko6klLIzkCr4cONXDJtUzKIPN/HHkb0Zf1I3t7CSJElqIAq65DNzfCF5OY0ZddcrzFq8NulISiGbaClj0epNDC2ay8atJTx0SX+G9O6QdCRJkiQdZF1b5zJzfCG9OuQx/uG/MmXOSmJ05W79fzbREvDvSz5h5JR5NG+azczxg+jXNT/pSJIkSUpIfm4THr6kP2f0as/Ns97hxqeWsqu0LOlYSglXSlKDd89f3uO3z77F8R1bcfeYvrRu0TTpSJIkSUpYs8bZ3DGqDx0PzWHynFWs2bSNO0b1IdfFZhs8Z6LVYJWWRX799FJueuYtTuvZjmmXDrCBliRJ0teysgLXnX40N519LLOXrefcKfNY/8X2pGMpYTbRapC2lZRy+UMLmVr8PmMLu1J0XgE5TbKTjiVJkqQUGj2gC3eP6cuqT79kaFExy9dtSTqSEmQTrQbn0y07GDllHi+8vY5fndmTG8/sSXaWK3BLkiRp7075TjumjxtISWkZwycVU7xiQ9KRlBCbaDUoK9ZvZdikuSxbt4XJ5xdwUWHXpCNJkiSpjujVMY8nJxTSPq8ZY+5bwOMLP0o6khJgE60GY/6qjQwrmsu2klKmjxvIacd8K+lIkiRJqmM6tMphxuWDOPGIfK6Z8Qa3v7DcLbAaGJtoNQhPLVrDBfcsoM0hTXlifCHHd2qVdCRJkiTVUXk5jZl6UT+Gn9CR2194l2tnvEnJLrfAaiga7PrsIYRmMUaX1qvnYoxMfGkFtzy/nP5d85kyui95zRsnHUuSJEl1XJNGWdwy4jg65zfntheWs3bzNiadX0Bejr9r1nepa6JDCEcDNwIrga7AJ8AvY4w79jEmB7geaAVsB74NXBdjXLqH93bMvPdw4Kwa/wSUGjtLy7j+iSVMf201Z/c+nH/+8XE0beQK3JIkSaoZIQSuPvXbdDw0h398/E1G3FnMvReeSMdDmycdTbUoVbdzhxB6AMXAozHG62OM51HeEE/bx5gsYAZwdIxxQozxGuB+YE4I4YhK7z0LuAG4DGhSK5+EUmHL9p2Mnfoq019bzVWndOO2c3vbQEuSJKlWDC/oyANj+7F283aGFhWzZM3mpCOpFqWqiQbuADbEGJ+qcOwWYGgI4Ud7GTMSOAO4dfeBGOPjwKbMWCocfxq4skYTK3XWbt7GiDvnUbxyI/88vBfXnNaDENzCSpIkSbVnULfWPH7FIJpkZ3HO5Hm8+M66pCOplqSmiQ4hdAK+B8ypdGoeUAJcupehY4EdwCuVjs8Bzg4htKl4MMa488DTKq2WfryZsyfO5aPPt3HfhSdy7omdk44kSZKkBqJ7u0N4YvwgjmyTyyX3v8aD8z9IOpJqQWqaaGBA5nVFxYOZpnc18PeVB4Ty6cX+wAcxxl2VTq8EsoFBNR9VafTy8k855855ZIXAjMsHMrh7m28eJEmSJNWgti2bMX3cQE7q0ZYbnlzCzbPepqzMLbDqkzQ10R0zrxv3cG4DcGgI4dBKx1sBLfYxBuDImomnNJu24EPGTn2Vzofl8sT4Qo5u3zLpSJIkSWqgcps2YsroAkYP6MKUOau4atrrbN9ZmnQs1ZA0rc6dk3nd0+3WJZnXXODzaoyplhDCOGAcQLt27Zg9e3Z1L1Xrtm7dmup8taUsRma+u5NnVu2kV+tsxh+zi3den887SQdrwBpqLSpdrEOlgXWotLAWk3NKXqSkRxOmL17LstXruPqEZhzSpGGu1VOf6jBNTfTumeM9bay2O+eXlY7vnoHenzFVFmOcAkwB6Nu3bzzppJOqe6laN3v2bNKcrzbs2FXKz2e8yTOrPmZUv078ZsixNM5O080VDVNDrEWlj3WoNLAOlRbWYrJOBr67eC0/nb6IW9+A+y46ka6tqz3PV2fVpzpMU8exNvNa+ZZtgMOAzTHGirPQZPaO/mwfYwDeq7GESo1NX5Uw+p4FPP3Gx/ziBz24eWgvG2hJkiSl0um92jPt0v58sX0Xw4rmsvCDz5KOpAOQpq7jVSACR1U8GEJoBHQGntvLuFeAI0IIlTcBPoryVbtn12xMJe3DjV8xbFIxiz7cxB9H9mb8Sd3cwkqSJEmpVtAln5lXDCIvpzGj7nqFZ99c+82DlEqpaaJjjJ8AL/K3q3D3A5oBDwOEEFpUWmDskcz5vpXGDQZmxRg31U5iJWHR6k0MLZrLxq0lPHRJf4b07pB0JEmSJKlKjmidy8zxhfTqkMeER/7KlDkridGVu+ua1DTRGdcCXUIIp1c6Ngd4JvPxQmBFCGH3gwTTgLnANbsHhBCGUL7a968q/wdCCC0y/3NPz1ErxZ5b+gkjp8yjedNsZo4fRL+u+UlHkiRJkvZLfm4THr6kP2f0as/Ns97hhqeWsKu0LOlY2g9pWliMGOOiEMJJwPWZ17bAGmB0jHF3Za2hfDXuXZkxpSGEHwK/DyHcR/kCZT2AwTHGxRWvH0L4PnBR5sNBIYRrgH+PMS6t3c9MB+rev7zHTc++xXEdW3HPmL60btE06UiSJElStTRrnM0do/rQ8dAcJs9ZxcebtnPHqD7kNk1Ve6a9SN3/SzHGBcBZ+zh/yh6ObQGurMK1n6P82eqRB5JRB09pWeSmZ95iavH7fP+Ydtx+bh9ymlR+/F2SJEmqW7KyAtedfjSd8ptz41NLOHfKPO4dcyJtWzZLOpq+Qdpu55a+tq2klCseWsjU4vcZW9iVovMKbKAlSZJUr5w/oAv3jDmRVZ9+ydCiYpav25J0JH0Dm2il0qdbdjByyjz+/PY6fnVmT248syfZWa7ALUmSpPrn5O+05U+XDaSktIzhk4opXrEh6UjaB5topc6K9VsZNmkuy9ZtYfL5BVxU2DXpSJIkSVKtOrZDHk9OKKR9XjMuuHcBjy38KOlI2gubaKXK/FUbGVY0l20lpTw6biCnHfOtpCNJkiRJB0WHVjnMuLx8F5prZ7zB7S8sdwusFLKJVmo8tWgNF9yzgDaHNOWJ8YX07tQq6UiSJEnSQZWX05ipF/Vj+Akduf2Fd/n5Y29SssstsNIkdatzq+GJMVI0eyV/eG4Z/bvmM2V0X/Kau423JEmSGqYmjbK4ZcRxdMrP4fYX3mXt5m0UnVdAXo6/I6eBM9FK1M7SMq6buZg/PLeMIb0P54GL+9lAS5IkqcELIfDTU7tz64jjeWXVZ4y4s5g1m7YlHUvYRCtBW7bvZOzUV3n01dVcdUo3bj+3N00buYWVJEmStNvwgo48MLYfazdv5+yJc1myZnPSkRo8m2glYu3mbYy4cx7FKzfyz8N7cc1pPQjBLawkSZKkygZ1a83jVwyiSXYW50yex4vvrEs6UoNmE62DbunHmzl74lw++nwb9114Iuee2DnpSJIkSVKqdW93CE+MH8SRbXK55P7XeHD+B0lHarBsonVQvbz8U865cx6BwIzLBzK4e5ukI0mSJEl1QtuWzZg+biAn9WjLDU8u4eZZb1NW5hZYB5tNtA6aRxd8yNipr9L5sFyenFDI0e1bJh1JkiRJqlNymzZiyugCzh/QmSlzVnHVtNfZvrM06VgNiltcqdaVlUVu/fMyJr60ku92b8PE806gRVNLT5IkSaqORtlZ3DTkWDrnN+fmWe/wyRfbueuCvuTnNkk6WoPgTLRq1Y5dpfx0+iImvrSSUf06cfeYvjbQkiRJ0gEKITBu8FEUnXcCi9dsZljRXN7f8GXSsRoEm2jVmk1flTD67gU8/cbH/OIHPbh5aC8aZ1tykiRJUk05vVd7pl3an83bdjK0aC4LP/gs6Uj1nh2NasWHG79i2KRiFq3exP8Z1YfxJ3VzCytJkiSpFhR0yeeJ8YXk5TRm1F2v8Oyba5OOVK/ZRKvGvf7h5wwtmsvGrSU8dEl/zjr+8KQjSZIkSfXaEa1zmTm+kF4d8pjwyF+ZMmclMbpyd22wiVaN+vclnzDqrvnkNm3EzPGD6Nc1P+lIkiRJUoOQn9uEhy/pzxm92nPzrHe44akl7CotSzpWveMKT6ox9/zlPX777Fsc37EVd4/pS+sWTZOOJEmSJDUozRpnc8eoPnQ8NIfJc1bx8abt3DGqD7ku7ltjnInWASsti/z66aXc9MxbnNazHdMuHWADLUmSJCUkKytw3elHc9PZxzJ72XrOnTKP9V9sTzpWvWETrQOyraSUyx9ayNTi9xlb2JWi8wrIaZKddCxJkiSpwRs9oAt3j+nLqk+/ZGhRMcvXbUk6Ur1gE61q+3TLDkZOmccLb6/j12f25MYze5Kd5QrckiRJUlqc8p12/OmygZSUljF8UjHFKzYkHanOs4lWtaxYv4WhRXNZtm4Lk88v4MLCrklHkiRJkrQHx3bI48kJhbTPa8YF9y7gsYUfJR2pTrOJ1n6bv2ojw4qK2b6zlOnjBnLaMd9KOpIkSZKkfejQKofHrhhE/yPzuXbGG9z+wnK3wKomm2jtlydfX8Poe16hbctmPDG+kOM7tUo6kiRJkqQqaNmsMfdd2I8fF3Tk9hfe5doZb1Kyyy2w9pfrnKtKYoxMfGkFtzy/nP5d85kyui95zRsnHUuSJEnSfmjSKIs//Pg4Oh3anNteWM7azduYdH4BeTn+bl9VzkTrG+0sLeOfHl/MLc8v5+zeh/PAxf1soCVJkqQ6KoTA1ad+m1tHHM+C9z5jxJ3FfPT5V0nHqjNsorVPW7bvZOzUV5n+2mquPLkbt53bm6aN3MJKkiRJquuGF3TkgbH9WLt5O0OLilmyZnPSkeoEm2jt1drN2xhx5zzmrdzIvww/jmu/34MQ3MJKkiRJqi8GdWvN41cMokl2FudMnseL76xLOlLq2URrj5Z+vJmzJ87lo8+3ce+FJ3LOiZ2SjiRJkiSpFnRvdwhPjB/EkW1yueT+13hw/gdJR0o1m2j9jZeXf8o5d84jKwRmXD6Qwd3bJB1JkiRJUi1q27IZ08cN5OQebbnhySXcPOttysrcAmtPbKL1Xzy64EPGTn2Vzofl8sT4Qo5u3zLpSJIkSZIOgtymjZg8uoDRA7owZc4qrpr2Ott3liYdK3Xc4koAlJVFbv3zMia+tJLvdm/DxPNOoEVTy0OSJElqSBplZ/GbIcfQOb85v5v1Np98sZ27LuhLfm6TpKOlhjPRYseuUn46fRETX1rJqH6duHtMXxtoSZIkqYEKIXDp4CMpOu8EFq/ZzLCiuby/4cukY6WGTXQDt+mrEkbfs4Cn3/iYn3+/BzcP7UXjbMtCkiRJauhO79WeaZf254vtuxhaNJeFH3yWdKRUsFtqwD7c+BXDJhWz6MNN/HFkbyac3M0trCRJkiR9raBLPjOvGEReTmNG3fUKsxavTTpS4myiG6hFqzcxbNJcNm4t4aFL+jOkd4ekI0mSJElKoSNa5zJzfCG9OuQx4ZG/ctecVcTYcFfutolugJ5b+gkjp8wjp0k2M8cPol/X/KQjSZIkSUqx/NwmPHxJf04/tj2/m/U2Nz61lF2lZUnHSoSrRzUw9/7lPW569i2O79iKu8f0pXWLpklHkiRJklQHNGuczR2j+tDx0Bwmz1nFx5u2ccdP+tC8ScNqK52JbiBKyyL/81+X8ptn3uK0nu2YdukAG2hJkiRJ+yUrK3Dd6Udz09nH8tKy9Zw7eT7rv9iedKyDyia6AdhWUsoVDy3kvrnvc/HfdaXovAJymmQnHUuSJElSHTV6QBfuHtOXlZ9uZWhRMcvXbUk60kFjE13Pbdi6g5F3zefPb6/jV2f25IYf9SQ7yxW4JUmSJB2YU77Tjj9dNpCS0jKGTyqmeOWGpCMdFDbR9diK9VsZWjSXZZ98weTzC7iosGvSkSRJkiTVI8d2yOPJCYW0z2vGmHsX8PjCj5KOVOtS9wR4COFo4EZgJdAV+AT4ZYxxxz7G5ADXA62A7cC3getijEsP9Np11SurNjLuwYU0zg5MHzeQ4zu1SjqSJEmSpHqoQ6scZlw+iCseWsg1M97go8+38d/+oRsh1M87YFPVRIcQegDFwIUxxqcyx54GpgHD9jImC5gBlMQYh2WODQfmhBAKYozvV/faddVTi9bw8xlv0ik/h6kX9aNTfvOkI0mSJEmqx/JyGjP1on5cN3Mxt72wnNWff8XNQ3vRpFH9u/k5bZ/RHcCG3U1uxi3A0BDCj/YyZiRwBnDr7gMxxseBTZmxB3LtOiXGyMSXVnD1o4vo07kVM68otIGWJEmSdFA0aZTFLSOO42enduexhR9x0dQFfLF9Z9KxalxqmugQQifge8CcSqfmASXApXsZOhbYAbxS6fgc4OwQQpsDuHadsassct3MxfzhuWWc3ftwHri4H3nNGycdS5IkSVIDEkLg6lO/za0jjueVVZ/x40nFrNm0LelYNSpNt3MPyLyuqHgwxrgzhLAa+PvKA0L5Tfb9gQ9ijLsqnV4JZAODgCb7e+26ZMv2ndz+1x0s2bCaq07pxn//Xvd6+/yBJEmSpPQbXtCR9nnNuOyhhZw9cS4Tjq0//UmamuiOmdeNezi3ATgqhHBojPHzCsdbAS32MQbgyArH9ufahBDGAeMA2rVrx+zZs7/xk0jCrPdKeGvjLsYe25SCJmt5+eW1SUdSA7Z169bUfq2o4bAOlQbWodLCWlSS/qmgMf974XamLimjS8uX6sVkX5qa6JzM655umi/JvOYCFRvdqo4pq8a1iTFOAaYA9O3bN5500kl7iZ6svx8c6fmvL3LhkH9IOorE7NmzSevXihoO61BpYB0qLaxFJe17393Of84t5uSTT046So1IUxO9e+Z4Tw/y7s75ZaXju2eWv2nMl1V8X52UnRU4Ii876RiSJEmS9DfatmzGYTmpWY7rgKWpid59D/Khezh3GLC58u3WMcYdIYTP9jEG4D2gdH+vLUmSJElSZWn6c8CrQASOqngwhNAI6Aw8t5dxrwBHhBAqT8UeRfmq3bMP4NqSJEmSJH0tNU10jPET4EX+dqXsfkAz4GGAEEKLEELFGeVHMuf7Vho3GJgVY9xU1WtLkiRJkrQvqWmiM64FuoQQTq90bA7wTObjhcCKEEJu5uNpwFzgmt0DQghDKF/t+1f7eW1JkiRJkvYqTc9EE2NcFEI4Cbg+89oWWAOMjjHuXmF7DeWrbO/KjCkNIfwQ+H0I4T7KFyjrAQyOMS7ez2tLkiRJkrRXqWqiAWKMC4Cz9nH+lD0c2wJceaDXliRJkiRpX9J2O7ckSZIkSallEy1JkiRJUhXZREuSJEmSVEU20ZIkSZIkVZFNtCRJkiRJVWQTLUmSJElSFYUYY9IZ6oQQwqfAB0nn2IfWlO+RLSXNWlQaWIdKA+tQaWEtKg3qQh12iTG2+aY32UTXEyGE12KMfZPOIVmLSgPrUGlgHSotrEWlQX2qQ2/nliRJkiSpimyiJUmSJEmqIpvo+mNK0gGkDGtRaWAdKg2sQ6WFtag0qDd16DPRkiRJkiRVkTPRkiRJkiRVUaOkA0iSJDU0IYSTgSHAWuD1GOPzCUeSpAMWQmgWY9yedI7aZhNdB4QQjgZuBFYCXYFPgF/GGHfsY0wOcD3QCtgOfBu4Lsa4tPYTqz6qZh0OAn4PnAh8DrwA/CLGuK72E6u+qk4tVhqfDRQDk2KMU2srp+q36tZhCCEfeJjyn8+jY4wrajur6q9q/mzuAvwjsBk4BOgB/DbG+HLtJ1Z9FULoSHnvcThwVhXefyjwW8r7lKZAO+BnMcaPajNnTfGZ6JQLIfQA5gMXxhifyhx7GtgVYxy2lzFZwNNAye73hBCGU/4wf0GM8f2DkV31RzXrsDvwPPAY5d8gzwB6A28CJ8YYSw5GdtUv1anFPVzj58C/ABfZRKs6qluHIYTDgP8EPgNObQizNao91fzZ3BJ4CxgTY/yPzLGBwH8AfWKMyw5KeNUrIYSzKP89bxzwXIzxB9/w/uaUfy+cE2P8WebYNcB4ynuVTbUc+YD5THT63QFs2P3NMeMWYGgI4Ud7GTOS8kK+dfeBGOPjwKbMWGl/VacOfwmcHGO8NsZ4PdAXeBE4Dvh+raZVfVadWvxa5o87A2ornBqM6tbhA0BnYJQNtGpAderwVKAD5c03ADHGecCXwA9rK6jqtxjj08CV+zHkp8AJVOhVgCLgW5TPZqeeTXSKhRA6Ad8D5lQ6NQ8oAS7dy9CxwA7glUrH5wBnhxDa1GRO1W/VqcMQQmPgjRjje7uPxRhLgbszHx5RK2FVrx3A98Td47OAX2f+SdVS3ToMIZwCnA7cH2NcXashVe8dwPfD3XeBnVzhWs0pv637kxqOqQYkxrhzP94+FlhV8dbtGOM2YAFwUQgh9Y8c20Sn2+7Zkv/yvFSmSFcDf195QAghAP2BD2KMuyqdXglkA4NqPqrqsf2uwxjjzhjjbXu41heZ1+U1mlANxX7XYiVXAg8CW2o+mhqQ6tbhhZnXXSGEF0IIm0IIH4YQrs/87Jb2R3Xr8D+ANcA9IYS+mWO/Af5M+eNXUq3KTOYdRaXazVgJ5APHHNRQ1WATnW4dM68b93BuA3Bo5qH8iloBLfYxBuDImomnBqI6dbg3RwMfUL7AmLS/ql2LIYQjgaNjjP9WW+HUYFS3Dgszr29TPoPYHngEuAn4RU2HVL1XrTrMzPb9iPIeoDiEUAysAobsYfJFqg3fVLtQB3oVm+h0y8m87un2iN234+TWwBhpX2qkpjIzLecDV2Vu7Zb2V7VqMVN7vwP+Ry3lUsNS3e+J7YG3Yox3xnLbgOson3m5LvMYjFRV1f7ZHGNcBPx3oAwYCFxG+cre0sFQL3oVm+h02/3XmD39YN39rMCXlY7v/qvO/oyR9qU6dbgn44BnY4z/WiOp1BBVtxYvBx6PMX5WK6nU0FS3DncB/2V7v1i+RcrzQB6uFaH9U+2fzSGEkcBFlC8wVkT5gp9/yTxnLdW2mvq9MlE20em2NvO6p9vCDgM2xxg/r3gwsy/gZ/sYA/DeHs5Je7PfdVhZCGEA5dtb1YkVF5Va+12LIYTOwN8Bz4QQmoUQmlG+HyVA48yx7FpLrPqout8T1/D/fw7v6XqtaiCbGo5q1WFmW6wHgX+KMW6MMU6gfL2IOrMqsuq8b6pdqAO9ik10ur0KRMofvv9aZsW6zsBzexn3CnDEHn4xPIryVbtn12xM1XPVrcPd7+sBjAGujG5MrwNTnVo8BfgJsK3Cv3cy56ZkPh5dS3lVP1X3e+I8oFNmlfiKdv9RJ/W/NCpVqluHZ1A+2/fG7gMxxonAX3HhWR0EMcYtlK8NcdQeTh9FeZP95kENVQ020SkWY/yE8n11K6+w2A9oBjwMEEJoUWnxiEcy5/tWGjcYmFUXNjBXehxAHe6eBfw5cHXF56BDCO1CCP1qNbjqnWrW4rOUa0ZCnQAAA7tJREFUP/NX8d+wzLnfZj5+tnaTqz45gO+JD1E+81JYaVx34JUY4wakKjqAOtz9zGnl7U5XAltrIapECOHQEEKLCoceAbqFEL5V4T1NKa/fR2OMZQc74/4KTgylWwihN1AM/DjGOCtzbCbltzucHGMsCyEsA1oDnWOMX2ZmoF8GPo4xnpMZMwSYBvSPMS5O4nNR3VXNOmxL+Q/4p4GvKlyuKfBDYHCMseJx6RtVpxb3cI0jKJ/1uyjGOPUgRVc9Ut06DCH8G+Wzh2fGGEsztfhG5uPK+/1K+3QAP5uXAPfFGP8xM6YV5TOD/xhjfCCJz0V1X6ZJ3gK8GGP8hwrHc4EPgU9jjN+p8N6/Ak/GGH+ROXY15QuAHh9jXFv5+mmT+o2sG7oY46IQwknA9ZnXtpQ/VzW6wl9p1lC+wt2uzJjSEMIPgd+HEO6j/AH+HpQ3LTbQ2m/7W4chhBzKF8s5hj3v9TfFBlrVUZ3viVJNO4A6HA78HvjXEMJ7wOHAMBtoVUc1f0dcn1mn5OYQwvTM+Y7A+BjjEwf5U1A9EUL4PuWL1QEMCiFcA/x7jHEp5bW3nvJaAyDGuDWEMBj4QwhhIuU12hEYVBcaaHAmWpIkSZKkKvOZaEmSJEmSqsgmWpIkSZKkKrKJliRJkiSpimyiJUmSJEmqIptoSZIkSZKqyCZakiR9LZSbHEJYE0KYHkLoHULoEkL4TdLZJElKA5toSZJU0RnAB5Tv+ZkHvA7MB+5PMpQkSWnhPtGSJOlrIYQ8YEuMsSzzcSvgyxjjzmSTSZKUDjbRkiRJkiRVkbdzS5IkSZJURTbRkiQJgBDCaSGEu0MIX4QQBmeOXRBCWB9COCfpfJIkpYG3c0uSpK+FEBoBHwPTgCeAbwNbgeIY4wdJZpMkKQ0aJR1AkiSlR4xxVwjhP4HBwKcxxt8mnUmSpDTxdm5JklTZK0Bv4O6kg0iSlDY20ZIkqbL3Mq+tE00hSVIK2URLkqSvhRCaAt8DdgGDEo4jSVLq2ERLkqSKfgr8FngDKAQIIRydaCJJklLEhcUkSWrgQgi9gH8APgHejzF+GEL4CzAyhHAGcAjwdpIZJUlKC2eiJUnSccBvgJ4xxumZY9OAHOC0GOOjiSWTJCll3CdakiRJkqQqciZakiRJkqQqsomWJEmSJKmKbKIlSZIkSaoim2hJkiRJkqrIJlqSJEmSpCqyiZYkSZIkqYpsoiVJkiRJqiKbaEmSJEmSqsgmWpIkSZKkKrKJliRJkiSpimyiJUmSJEmqov8H0MKbqGYtPL4AAAAASUVORK5CYII=\n",
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