USAJMO 2018 Question 3

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $AC\perp BD$. Let $E,F$ be the reflections of $D$ across $BA,BC$, and let $P=BD\cap EF$. The circumcircles of $EPD,FPD$ meet $\omega$ again at $Q,R$, respectively, different from $D$. Prove that $EQ=FR$.

Proposed by Ray Li.