TSTST 2020 Problem 2

Let $A B C$ be a scalene triangle with incenter $I$. The incircle of $A B C$ touches $\overline{B C}$, $\overline{C A}, \overline{A B}$ at points $D, E, F$, respectively. Let $P$ be the foot of the altitude from $D$ to $\overline{E F}$, and let $M$ be the midpoint of $\overline{B C}$. The rays $A P$ and $I P$ intersect the circumcircle of triangle $A B C$ again at points $G$ and $Q$, respectively. Show that the incenter of triangle $G Q M$ coincides with $D$.