Let $S$ be a set of $31$ equally spaced points on a circle centered at $O$, and consider a uniformly random pair of distinct points $A$ and $B$ ($A, B \in S$). The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed as $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$.

This problem is posed by Muhammad A.

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