PMO 2019 Problem 4

In acute triangle $ABC$ with $\angle BAC > \angle BCA$ , let $P$ be the point on side $BC$ such that $\angle PAB = \angle BCA$ . The circumcircle of triangle $AP B$ meets side $AC$ again at $Q$ . Point $D$ lies on segment $AP$ such that $\angle QDC = \angle CAP$ . Point $E$ lies on line $BD$ such that $CE = CD$ . The circumcircle of triangle $CQE$ meets segment $CD$ again at $F$ , and line $QF$ meets side $BC$ at $G$ . Show that $B, D, F$ , and $G$ are concyclic