Point $H$ is the orthocenter of triangle $ABC$. Points $D, E$ and $F$ lie on the circumcircle of triangle $ABC$ such that $AD\parallel BE\parallel CF$. Points $S, T,$ and $U$ are the respective reflections of $D, E, F$ across the lines $BC, CA$ and $AB$. Prove that $S, T, U, H$ are cyclic.