CN-TST 2019 Problem 7

$AB$ and $AC$ are tangents to a circle $\omega$ with center $O$ at $B,C$ respectively. Point $P$ is a variable point on minor arc $BC$ . The tangent at $P$ to $\omega$ meets $AB,AC$ at $D,E$ respectively. $AO$ meets $BP,CP$ at $U,V$ respectively. The line through $P$ perpendicular to $AB$ intersects $DV$ at $M$ , and the line through $P$ perpendicular to $AC$ intersects $EU$ at $N$ . Prove that as $P$ varies, $MN$ passes through a fixed point.