2024 AMC 12B Problem 9

A dartboard is the region B in the coordinate plane consisting of points (x,y) such that |x| + |y| \le 8 . A target T is the region where (x^2 + y^2 - 25)^2 \le 49. A dart is thrown and lands at a random point in B. The probability that the dart lands in T can be expressed as \frac{m}{n} \cdot \pi, where m and n are relatively prime positive integers. What is m + n?

\textbf{(A) }39 \qquad \textbf{(B) }71 \qquad \textbf{(C) }73 \qquad \textbf{(D) }75 \qquad \textbf{(E) }135 \qquad

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