Advanced Number Theory · Superchallenge problem

In triangle $ABC$, cevians $AD$, $BE$, and $CF$ are concurrent at a point $P$, where $D$, $E$, $F$ lie on sides $BC$, $CA$, $AB$ respectively, with $BD:DC = 2:3$ and $CE:EA = 1:2$. Then $AF:FB = m:n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.