# 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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