TxO Math Bowl 2024 - Guts Contest - Set 4 Problem 3

For all positive integers n, let the n^{\text{th}} golden circle be the circle with diameter (0,0) to \left(\frac{1}{2^{n-1}}, 0\right). If the area of the region of points that are enclosed within an odd number of golden circles can be written as \frac{m\pi}{n}, where m and n are relatively prime positive integers, find m + n.