BxMO 2024 Problem 3

Let $A B C$ be a triangle with incentre $I$ and circumcircle $\Omega$ such that $|A C| \neq|B C|$. The internal angle bisector of $\angle C A B$ intersects side [BC] in $D$, and the external angle bisectors of $\angle A B C$ and $\angle B C A$ intersect $\Omega$ again in $E$ and $F$, respectively. Let $G$ be the intersection of lines $A E$ and $F I$ and let $\Gamma$ be the circumcircle of triangle $B D I$. Show that $E$ lies on $\Gamma$ if and only if $G$ lies on $\Gamma$.