2024 AMC 10A Problem 14

One side of an equilateral triangle of height 24 lies on line \ell. A circle of radius 12 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?

\textbf{(A)}~72\qquad\textbf{(B)}~73\qquad\textbf{(C)}~74\qquad\textbf{(D)}~75\qquad\textbf{(E)}~76

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.