John's Red and Blue Balls

Probability Level 5

John has a box that contains one red ball and one blue ball. Every minute he reaches into the box and randomly takes out a ball from it (without looking). No matter which ball he took out, he puts it back together with{together\ with} another ball of the same color. Thus, after nn minutes, the box contains a total of n+2n+2 balls.

The probability that after 22 hours, the box contains exactly 7777 blue balls, can be written as ab\frac{a}{b}, where aa and bb are coprime positive integers. Find a+ba+b.

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