HMMT 2022 G10

Suppose $\omega$ is a circle centered at $O$ with radius 8 . Let $A C$ and $B D$ be perpendicular chords of $\omega$. Let $P$ be a point inside quadrilateral $A B C D$ such that the circumcircles of triangles $A B P$ and $C D P$ are tangent, and the circumcircles of triangles $A D P$ and $B C P$ are tangent. If $A C=2 \sqrt{61}$ and $B D=6 \sqrt{7}$, then $O P$ can be expressed as $\sqrt{a}-\sqrt{b}$ for positive integers $a$ and $b$. Compute $100 a+b$.