PMO 2023 Problem 3

In $\triangle ABC$ , $AB > AC$ . Point $P$ is on line $BC$ such that $AP$ is tangent to its circumcircle. Let $M$ be the midpoint of $AB$ , and suppose the circumcircle of $\triangle PMA$ meets line $AC$ again at $N$ . Point $Q$ is the reflection of $P$ with respect to the midpoint of segment $BC$ . The line through $B$ parallel to $QN$ meets $PN$ at $D$ , and the line through $P$ parallel to $DM$ meets the circumcircle of $\triangle PMB$ again at $E$ . Show that the lines $PM$ , $BE$ , and $AC$ are concurrent.