Silk Road Mathematics Competition

Let $A$, $B$, $C$ be three collinear points such that the point $B$ lies between $A$ and $C$. Let $A{A}^{\prime }$ and $B{B}^{\prime }$ be parallel lines such that the points $A^{\prime }$ and $B^{\prime }$ lie on the same side of the line $AB$, and $A^{\prime }$, $B^{\prime }$, $C$ are not collinear. Let $O_1$ be the center of the circle passing through the points $A$, $A^{\prime }$, $C$, and $O_2$ be the center of the circle passing through the points $B$, $B^{\prime }$, $C$. Find all possible values of the angle $CA{A}^{\prime }$, if triangles $A^{\prime }C{B}^{\prime }$ and $O_{1}C{O}_2$ have the same area.