AMC 12 2020b Problem 10

In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$

$\textbf{(A) } \frac{\sqrt5}{12} \textbf{(B) } \frac{\sqrt5}{10} \textbf{(C) } \frac{\sqrt5}{9} \textbf{(D) } \frac{\sqrt5}{8} \textbf{(E) } \frac{2\sqrt5}{15}$