KöMaL 2022 C1700

In a circle of centre $O$, $A$ is an interior point different from $O$. For a point $B$ on the circumference of the circle, $\angle OAB =\alpha$. Let $C$ be a point on the circumference such that $2\alpha+\beta=180^{\circ}$, where $\angle BAC =\beta$, and the angles $\angle BAO$ and $\angle BAC$ have no common points apart from the ray $AB$. Prove that the points $O$, $A$, $B$, $C$ are concyclic. \