BALKAN-SL 2021 G8

Let $A B C$ be a scalene triangle and let $I$ be its incenter. The projections of $I$ on $B C, C A$ and $A B$ are $D, E$ and $F$ respectively. Let $K$ be the reflection of $D$ over the line $A I$, and let $L$ be the second point of intersection of the circumcircles of the triangles $B F K$ and $C E K$. If $\frac{1}{3} B C=A C-A B$, prove that $D E=2 K L$.

Proposed by Romania