2024 AMC 10B Problem 14

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 \dfrac{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

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