US-EGMO-TST 2020 Problem 4

Let $ABC$ be a triangle. Distinct points $D$ , $E$ , $F$ lie on sides $BC$ , $AC$ , and $AB$ , respectively, such that quadrilaterals $ABDE$ and $ACDF$ are cyclic. Line $AD$ meets the circumcircle of $\triangle ABC$ again at $P$ . Let $Q$ denote the reflection of $P$ across $BC$ . Show that $Q$ lies on the circumcircle of $\triangle AEF$ .

Proposed by Ankan Bhattacharya