Two players \(A\) and \(B\) alternately throw two fair six-sided dice. Player \(A\) wins if he scores 6 points before \(B\) scores 7 points. Otherwise, player \(B\) wins. Calculate the probability that \(A\) wins, knowing that this player starts the game.

If the value of this probability can be written as \(\displaystyle{\frac{m}{n}}\), where \(m\) and \(n\) are relatively prime integers, find \(m+n\).

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