Distribution on a grid

Discrete Mathematics Level pending

A token is randomly placed into a square on a \(14 \times 12\) chessboard according to a probability distribution \(P.\) The token is then moved uniformly at random to one of the horizontally, vertically, or diagonally adjacent squares. The probability that the token is in a particular position after it has been moved also satisfies the distribution \(P.\) The token is again moved uniformly at random to an adjacent square. Let \(q_P\) be the probability that the first and last position of the token is the same for the distribution \(P\) and let \(q\) be the maximum value of \(q_P\) over all distributions \(P\) which satisfy the above relation. \(q\) can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b?\)

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