JBMO 2023 Problem 4

Let $A B C$ be an acute triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $B C$ and let $M$ be the midpoint of $O D$. The points $O_{b}$ and $O_{c}$ are the circumcenters of triangles $A O C$ and $A O B$, respectively. If $A O=A D$, prove that the points $A, O_{b}, M$ and $O_{c}$ are concyclic.