Advanced Number Theory · Superchallenge problem

In triangle $ABC$, point $D$ lies on $BC$ with $BD:DC = 1:2$ and point $E$ lies on $CA$ with $CE:EA = 1:2$. Cevians $AD$ and $BE$ intersect at $P$. Then $\dfrac{AP}{PD} = \dfrac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.