Circles $\omega$ and $\gamma$, both centered at $O$, have radii $26$ and $14$, 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 $\frac{BP}{CP}=4$. Then $AB$ can be written in the form $\frac{m}{\sqrt{n}}-\frac{p}{\sqrt{q}}$ for positive integers $m,n,p,q$ with $\gcd (m,n)=\gcd (p,q)=1$. What is $m+n+p+q$?