A B C D is a square of side length \sqrt{3}+1. Point P is on \overline{A C} such that A P=\sqrt{2}. The square region bounded by A B C D is rotated 90^{\circ} counterclockwise with center P, sweeping out a region whose area is \dfrac{1}{c}(a \pi+b), where a, b, and c are positive integers and \operatorname{gcd}(a, b, c)=1. What is a+b+c ?
Answer Choices
A. 15
B. 17
C. 19
D. 21
E. 23