Brilli the ant stands at a vertex, \(A\), of an icosahedron. After every minute, he travels to one of the five adjacent vertices, along an edge, with equal probability. After \(10\) minutes, having traveled \(10\) times, the probability that he is back at vertex \(A\) can be expressed as \(\frac{m}{5^{10}}\). Find the value of \(m\)

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