One side of an equilateral triangle of height $30$ lies on line $\ell$. A circle of radius $6\sqrt{6}$ is tangent to line $\ell$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a\sqrt{b}-c\pi$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c$?