ELMO SL 2019 G1

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$ . Let $BH$ intersect $AC$ at $E$ , and let $CH$ intersect $AB$ at $F$ . Let $AH$ intersect $\Gamma$ again at $P \neq A$ . Let $PE$ intersect $\Gamma$ again at $Q \neq P$ . Prove that $BQ$ bisects segment $\overline{EF}$ .

Proposed by Luke Robitaille