In the figure shown below, major arc $\widehat{AD}$ and minor arc $\widehat{BC}$ have the same center, $O$. Also, $A$ lies between $O$ and $B$, and $D$ lies between $O$ and $C$. Major arc $\widehat{AD}$, minor arc $\widehat{BC}$, and each of the two segments $\overline{AB}$ and $\overline{CD}$ have length $2\pi$. What is the distance from $O$ to $A$? $\textbf{(A)}~1 \qquad$ $ \textbf{(B)}~1 - \pi + \sqrt{\pi^{2} + 1} \qquad$ $ \textbf{(C)}~\frac{\pi}{2} \qquad$ $ \textbf{(D)}~\frac{\sqrt{\pi^{2} + 1}}{2} \qquad$ $ \textbf{(E)}~2$