BALTIC-WAY 2024 Problem 14

Let $A B C$ be an acute triangle with circumcircle $\omega$. The altitudes $A D, B E$ and $C F$ of the triangle $A B C$ intersect at point $H$. A point $K$ is chosen on the line $E F$ such that $K H \| B C$. Prove that the reflection of $H$ in $K D$ lies on $\omega$.