BALTIC-WAY 2023 Problem 11

$$ Let $A B C$ be a triangle and let $J$ be the centre of the $A$-excircle. The reflection of $J$ in $B C$ is $K$. The points $E$ and $F$ are on $B J$ and $C J$, respectively, such that $\angle E A B=\angle C A F=90^{\circ}$. Prove that $\angle F K E+\angle F J E=$ $180^{\circ}$.

Remark: The $A$-excircle is the circle that touches the side $B C$ and the extensions of $A C$ and $A B$.