Advanced Number Theory · Superchallenge problem

In triangle $ABC$, $D$ is the midpoint of $BC$, and $E$ lies on $CA$ with $CE:EA = 1:3$. Cevians $AD$, $BE$, and $CF$ (with $F$ on $AB$) are concurrent. Line $DF$ is extended to meet line $AC$ at $X$. Then $\dfrac{AX}{XC} = \dfrac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.