Mongolian Mathematical Olympiad

Let $ABC$ be an acute triangle, where the altitudes $BD$ and $CE$ are drawn. Let $L$ be the point on segment $BD$ such that $AD=DL$, and let $K$ be the point on segment $CE$ such that $AE=EK$. Denote $M$ as the midpoint of segment $KL$. The circumcircle of triangle $ABC$ intersects line $AL$ again at point $T$ and line $AK$ again at point $S$. Prove that the lines $BS$, $CT$, and $AM$ are concurrent.