Given a circle centered at point $O$, with $AB$ as the diameter. Point $C$ lies on the extension of line $AB$ so that $B$ lies between $A$ and $C$, and the line through $C$ intersects the circle at points $D$ and $E$ (where $D$ lies between $C$ and $E$). $OF$ is the diameter of the circumcircle of triangle $OBD$, and the extension of the line $CF$ intersects the circumcircle of triangle $OBD$ at point $G$. Prove that the points $O, A, E, G$ lie on a circle.