The figure shows a network of roads bounding \(12\) blocks. Person \(P\) goes from \(A\) to \(B,\) and person \(Q\) goes from \(B\) to \(A,\) each going by a shortest path (along roads). The persons start simultaneously and go at the same constant speed. At each point with two possible directions to take, both have the same probability. The probability that the persons meet can be expressed as \(\frac{A}{B}\) where \(A\) and \(B\) are positive coprime integers. Find \(A+B\).
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