Advanced Number Theory · Superchallenge problem

Square $ABCD$ is inscribed in a circle, and $P$ lies on arc $CD$ (the arc not containing $A$ or $B$). Given $PC=3$ and $PD=4$, the length $PA$ can be written $a+b\sqrt{c}$ with $a$, $b$, $c$ positive integers and $c$ squarefree. Find $a+b+c$.