$ABCD$ is a square of side length $2\sqrt{3}+2$. Point $P$ is on $\overline{AC}$ such that $AP=2\sqrt{2}$. 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 $a,b,c$ are positive integers and $\gcd (a,b,c)=1$. What is $a+b+c$?