HMMT 2016 T10

Let $A B C$ be a triangle with incenter $I$ whose incircle is tangent to $\overline{B C}, \overline{C A}, \overline{A B}$ at $D, E, F$. Point $P$ lies on $\overline{E F}$ such that $\overline{D P} \perp \overline{E F}$. Ray $B P$ meets $\overline{A C}$ at $Y$ and ray $C P$ meets $\overline{A B}$ at $Z$. Point $Q$ is selected on the circumcircle of $\triangle A Y Z$ so that $\overline{A Q} \perp \overline{B C}$. Prove that $P, I, Q$ are collinear.