Advanced Number Theory · Superchallenge problem

In triangle $ABC$, cevians $AD$, $BE$, $CF$ are concurrent, with $F$ on $AB$ satisfying $AF:FB = 1:2$ and $D$ on $BC$ satisfying $BD:DC = 3:4$. Then $CE:EA = \tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.