# Token on a grid

Brilli the ant randomly placed a token into a square on a $$2 \times 100$$ 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 $$P.$$ Let $$q$$ be the probability that the token is placed into one of the columns of $$C = \{5,6, \ldots 44\}$$ and after being moved is still in one of those columns. The value of $$q$$ can be expressed as $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a + b?$$

Details and assumptions

$$C$$ is the set of integers from 5 to 44 (inclusive).

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