SHARYGIN 2023 Problem 10

Altitudes $BE$ and $CF$ of an acute-angled triangle $ABC$ meet at point $H$ . The perpendicular from $H$ to $EF$ meets the line $\ell$ passing through $A$ and parallel to $BC$ at point $P$ . The bisectors of two angles between $\ell$ and $HP$ meet $BC$ at points $S$ and $T$ . Prove that the circumcircles of triangles $ABC$ and $PST$ are tangent.