Advanced Number Theory · Superchallenge problem

In triangle $ABC$, point $D$ lies on $BC$ with $BD:DC = 2:3$ and point $F$ lies on $AB$ with $AF:FB = 3:4$. Cevians $AD$ and $CF$ intersect at $P$. Then $\dfrac{AP}{PD} = \dfrac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.