2025 AMC 12A Problem 10

In the figure shown below, major arc AD and minor arc BC have the same center, O. Also, A lies between O and B, and D lies between O and C. Major arc AD, minor arc BC, and each of the two segments AB and CD have length 2\pi.

What is the distance from O to A?

\textbf{(A)} 1 \qquad \textbf{(B)} 1 - \pi + \sqrt{1 + \pi^2} \qquad \textbf{(C)} \dfrac{1}{2}\pi \qquad \textbf{(D)} \dfrac{1}{2}\sqrt{1 + \pi^2} \qquad \textbf{(E)} 2

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