INMO 2024 Problem 1

In triangle $A B C$ with $C A=C B$, point $E$ lies on the circumcircle of $A B C$ such that $\angle E C B=90^{\circ}$. The line through $E$ parallel to $C B$ intersects $C A$ in $F$ and $A B$ in $G$. Prove that the centre of the circumcircle of triangle $E G B$ lies on the circumcircle of triangle $E C F$.