Mongolian Mathematical Olympiad

In a triangle $KLM$, $KM=6$ and let $LP$ be a median of the $KLM$. Let $P_1$, $P_2$ be bases of perpendiculars dropped from $P$ to sides $KL$, $LM$ respectively. If $P{P}_1:P{P}_2=1:2$ and area of the triangle $KLM$ is maximal then find the length of the median $LP$.