Advanced Number Theory · Challenge problem

Square $ABCD$ has side $12$. Points $E$ and $F$ lie on sides $\overline{BC}$ and $\overline{CD}$ respectively so that $\angle EAF = 45^\circ$. The smallest possible length of $\overline{EF}$ is $a\sqrt b - c$ for positive integers $a$, $b$, $c$ with $b$ squarefree. Find $a + b + c$.