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 $\frac{BP}{CP}=3$. Find the length of  $AB$.