Fantastic Ratios

Let $ABCD$ be a square and $N$ a point on segment $CD$ such that $AD = 3 ND$ . Let $M$ be a point on segment $BC$ satisfying $\angle MAN = 45^\circ$ . The ratio $\frac{AM}{AN}$ can be expressed as $\frac{a\sqrt{b}}{c}$ , where $a, b,$ and $c$ are positive integers, $a$ and $b$ are relatively prime, and $c$ is squarefree. Find the value of $ac - b^2$ .