2025 AMC 10A Problem 13

In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is k, where 0 \lt k \lt 1. The spaces between squares are alternately shaded, as shown in the figure (which is not necessarily drawn to scale).

The area of the shaded portion of the figure is 64\% of the area of the original square. What is k?

\textbf{(A)} \dfrac{3}{5} \qquad \textbf{(B)} \dfrac{16}{25} \qquad \textbf{(C)} \dfrac{2}{3} \qquad \textbf{(D)} \dfrac{3}{4} \qquad \textbf{(E)} \dfrac{4}{5}

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