BALTIC-WAY 2021 Problem 12

Let $I$ be the incentre of a triangle $A B C$. Let $F$ and $G$ be the projections of $A$ onto the lines $B I$ and $C I$, respectively. Rays $A F$ and $A G$ intersect the circumcircles of the triangles $C F I$ and $B G I$ for the second time at points $K$ and $L$, respectively. Prove that the line $A I$ bisects the segment $K L$.