HMMT November 2020 U20

Let $\omega_{1}$ be a circle of radius 5 , and let $\omega_{2}$ be a circle of radius 2 whose center lies on $\omega_{1}$. Let the two circles intersect at $A$ and $B$, and let the tangents to $\omega_{2}$ at $A$ and $B$ intersect at $P$. If the area of $\triangle A B P$ can be expressed as $\frac{a \sqrt{b}}{c}$, where $b$ is square-free and $a, c$ are relatively prime positive integers, compute $100 a+10 b+c$.