CZ-SK-MO 2022 Problem 3

Given a scalene acute triangle $ABC$ , let M be the midpoints of its side $BC$ and $N$ the midpoint of the arc $BAC$ of its circumcircle. Let $\omega$ be the circle with diameter $BC$ and $D$ , $E$ its intersections with the bisector of angle $\angle BAC$ . Points $D'$ , $E'$ lie on $\omega$ such that $DED'E'$ is a rectangle. Prove that $D'$ , $E'$ , $M$ , $N$ lie on a single circle.

(Patrik Bak)