2014 Pascal Problem 20

In the diagram, line segment PS has length 4. Points Q and R are on line segment PS. Four semi-circles are drawn on the same side of PS. The diameters of these semi-circles are PS, PQ, QR, and RS. The region inside the largest semi-circle and outside the three smaller semi-circles is shaded. What is the area of a square whose perimeter equals the perimeter of the shaded region?

\textbf{(A)}\ 4\quad \textbf{(B)}\ \pi\quad \textbf{(C)}\ \pi^2\quad \textbf{(D)}\ 2\pi^2\quad \textbf{(E)}\ \frac{\pi^2}{4}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Pascal contests here.