A token is placed at one of 9 positions in a $3 \times 3$ grid according to a probability distribution $P.$ After a token is placed into one of the positions of the grid, it is then moved uniformly at random to one of the horizontally, vertically, or diagonally adjacent positions. For each position on the board, the probability that the token is in that position after being moved is also given by the distribution $P.$

If two tokens are placed into the grid according to the distribution $P$ and then moved, the probability that the set of occupied positions is the same before and after the tokens are moved 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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