AMC 10 2008a Problem 19

Rectangle $PQRS$ lies in a plane with $PQ=RS=2$ and $QR=SP=6$. The rectangle is rotated $90^\circ$ clockwise about $R$, then rotated $90^\circ$ clockwise about the point $S$ moved to after the first rotation. What is the length of the path traveled by point $P$? $\mathrm{(A)}\ \left(2\sqrt{3}+\sqrt{5}\right)\pi \mathrm{(B)}\ 6\pi \mathrm{(C)}\ \left(3+\sqrt{10}\right)\pi \mathrm{(D)}\ \left(\sqrt{3}+2\sqrt{5}\right)\pi\\\mathrm{(E)}\ 2\sqrt{10}\pi$