US-TST 2018 Problem 5

Let $A B C D$ be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at $H$. Denote by $M$ and $N$ the midpoints of $\overline{B C}$ and $\overline{C D}$. Rays $M H$ and $N H$ meet $\overline{A D}$ and $\overline{A B}$ at $S$ and $T$, respectively. Prove there exists a point $E$, lying outside quadrilateral $A B C D$, such that - ray $E H$ bisects both angles $\angle B E S, \angle T E D$, and - $\angle B E N=\angle M E D$.