Circles \omega and \gamma, both centered at O, have radii 20 and 17 , respectively. Equilateral triangle A B C, 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{B C} is tangent to \gamma. Segments \overline{A O} and \overline{B C} intersect at P, and \dfrac{B P}{C P}=3. Then A B can be written in the form \dfrac{m}{\sqrt{n}}-\dfrac{p}{\sqrt{q}} for positive integers m, n, p, q with \operatorname{gcd}(m, n)=\operatorname{gcd}(p, q)=1. What is m+n+p+q ?
Answer Choices
A. 42
B. 86
C. 92
D. 114
E. 130