JBMO 2022 Problem 2

Let $A B C$ be an acute triangle such that $A H=H D$, where $H$ is the orthocenter of $A B C$ and $D \in B C$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $B H C$. Let $S$ and $T$ be the intersection points of $\ell$ with $A B$ and $A C$, respectively. Denote the midpoints of $B H$ and $C H$ by $M$ and $N$, respectively. Prove that the lines $S M$ and $T N$ are parallel.