IGO 2020 E4

Let $P$ be an arbitrary point in the interior of triangle $\triangle ABC$ . Lines $\overline{BP}$ and $\overline{CP}$ intersect $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$ , respectively. Let $K$ and $L$ be the midpoints of the segments $BF$ and $CE$ , respectively. Let the lines through $L$ and $K$ parallel to $\overline{CF}$ and $\overline{BE}$ intersect $\overline{BC}$ at $S$ and $T$ , respectively; moreover, denote by $M$ and $N$ the reflection of $S$ and $T$ over the points $L$ and $K$ , respectively. Prove that as $P$ moves in the interior of triangle $\triangle ABC$ , line $\overline{MN}$ passes through a fixed point. Proposed by Ali Zamani