AMC 12 2012a 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} \textbf{(B)}\ \frac{2\sqrt{5}}{5} \textbf{(C)}\ \frac{2\sqrt{2}}{3} \textbf{(D)}\ \frac{2\sqrt{19}-4}{5} \textbf{(E)}\ \frac{9-\sqrt{17}}{5}