Question 10
A circle with center $O$ has radius $25.$ Chord $\overline{AB}$ of length $30$ and chord $\overline{CD}$ of length $14$ intersect at point $P.$ The distance between the midpoints of the two chords is $12.$ The quantity $OP^2$ can be represented as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m + n$ is divided by $1000.$