2013 AMC 12A Problem 23

ABCD is a square of side length \sqrt{3} + 1. Point P is on \overline{AC} such that AP = \sqrt{2}. The square region bounded by ABCD is rotated 90^{\circ} counterclockwise with center P, sweeping out a region whose area is \frac{1}{c} (a \pi + b), where a, b, and c are positive integers and \text{gcd}(a,b,c) = 1. What is a + b + c?

\textbf{(A)} \ 15 \qquad \textbf{(B)} \ 17 \qquad \textbf{(C)} \ 19 \qquad \textbf{(D)} \ 21 \qquad \textbf{(E)} \ 23

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