Probability of a Mutual Meeting!

The figure above shows a street plan of \(12\) square blocks. A person \(P\) goes from the point \(A\) to the point \(B\), and a second person \(Q\) goes from \(B\) to \(A\). Both of them (\(P\) and \(Q\)) leave at the same time with the same speed, following shortest paths on the grid. At each corner, they choose among the possible streets with equal probability.

If the probability that \(P\) meets \(Q\) can be expressed as \(\dfrac{\alpha}{\beta}\) where \(\alpha,\beta \in \mathbb N\), find the minimum value of \(\alpha + \beta\).

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