SHARYGIN 2018 Problem 4

Let $ABCD$ be a cyclic quadrilateral. A point $P$ moves along the arc $AD$ which does not contain $B$ and $C$. A fixed line $l$, perpendicular to $BC$, meets the rays $BP, CP$ at points $B_0, C_0$ respectively. Prove that the tangent at $P$ to the circumcircle of triangle $PB_0C_0$ passes through some fixed point.