BALTIC-WAY 2021 Problem 14

Let $A B C$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. Denote by $M$ the midpoint of $B C$. The point $D$ is the reflection of $A$ over $B C$, and the point $E$ is the intersection of $\Gamma$ and the ray $M D$. Let $S$ be the circumcentre of the triangle $A D E$. Prove that the points $A, E, M, O$, and $S$ lie on the same circle.