IMSO 2015 — Math Division — Essay Question 12

In triangle $ABC$, points $D$ and $E$ are on $BC$ such that $BD:DE:EC=2:1:1$. If point $M$ is on $AC$ such that $\frac{CM}{MA}=\frac13$ and $BM$ intersects $AD$ and $AE$ at points $H$ and $G$ respectively. Find $BH:HG:GM$.