Given a triangle $ABC$ with $M$ the midpoint of minor arc $BC$. Let $H$ be the feet of altitude from $A$ to $BC$. Let $S$ and $T$ be the reflections of $B$ and $C$ with respect to line $AM$. Suppose the circle $(HST)$ meets $BC$ again at a point $P$. Prove that $\angle AMP = 90^\circ$. (Proposed by Tan Rui Xuen)