PAMO 2023 Problem 6

Let $ABC$ be an acute triangle with $AB<AC$ . Let $D, E,$ and $F$ be the feet of the perpendiculars from $A, B,$ and $C$ to the opposite sides, respectively. Let $P$ be the foot of the perpendicular from $F$ to line $DE$ . Line $FP$ and the circumcircle of triangle $BDF$ meet again at $Q$ . Show that $\angle PBQ = \angle PAD$ .