2012 AMC 12A Problem 12

A square region ABCD is externally tangent to the circle with equation x^2+y^2=1 at the point (0,1) on the side CD. Vertices A and B are on the circle with equation x^2+y^2=4. What is the side length of this square?

\textbf{(A)}\ \frac{\sqrt{10}+5}{10}\qquad\textbf{(B)}\ \frac{2\sqrt{5}}{5}\qquad\textbf{(C)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(D)}\ \frac{2\sqrt{19}-4}{5}\qquad\textbf{(E)}\ \frac{9-\sqrt{17}}{5}

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