2019 AMC 12A Problem 22

Circles \omega and \gamma, both centered at O, have radii 20 and 17, respectively. Equilateral triangle ABC, whose interior lies in the interior of \omega but in the exterior of \gamma, has vertex A on \omega, and the line containing side \overline{BC} is tangent to \gamma. Segments \overline{AO} and \overline{BC} intersect at P, and \dfrac{BP}{CP} = 3. Then AB can be written in the form \dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}} for positive integers m, n, p, q with \gcd(m,n) = \gcd(p,q) = 1. What is m+n+p+q? \phantom{ }

\textbf{(A) } 42 \qquad \textbf{(B) }86 \qquad \textbf{(C) } 92 \qquad \textbf{(D) } 114 \qquad \textbf{(E) } 130

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.