HMMT 2022 T5

Let $A B C$ be a triangle with centroid $G$, and let $E$ and $F$ be points on side $B C$ such that $B E=E F=F C$. Points $X$ and $Y$ lie on lines $A B$ and $A C$, respectively, so that $X, Y$, and $G$ are not collinear. If the line through $E$ parallel to $X G$ and the line through $F$ parallel to $Y G$ intersect at $P \neq G$, prove that $G P$ passes through the midpoint of $X Y$.