UA-UMO 2023 Problem 93

You are given an acute triangle $ABC$ with circumcircle $\omega$ . Points $F$ on $AC$ , $E$ on $AB$ and $P, Q$ on $\omega$ are chosen so that $\angle AFB = \angle AEC = \angle APE = \angle AQF = 90^\circ$ . Show that lines $BC, EF, PQ$ are concurrent or parallel.

Proposed by Fedir Yudin