SHARYGIN 2022 Problem 11

Let $ABC$ be a triangle with $\angle A=60^o$ and $T$ be a point such that $\angle ATB=\angle BTC=\angle ATC$ . A circle passing through $B,C$ and $T$ meets $AB$ and $AC$ for the second time at points $K$ and $L$ .Prove that the distances from $K$ and $L$ to $AT$ are equal.