HMMT 2025 T8

Let $\triangle ABC$ be a triangle with incenter $I$ . The incircle of triangle $\triangle ABC$ touches $\overline{BC}$ at $D$ . Let $M$ be the midpoint of $\overline{BC}$ , and let line $AI$ meet the circumcircle of triangle $\triangle ABC$ again at $L \neq A$ . Let $\omega$ be the circle centered at $L$ tangent to $AB$ and $AC$ . If $\omega$ intersects segment $\overline{AD}$ at point $P$ , prove that $\angle IPM = 90^{\circ}$ .