Stewart's Theorem

In triangle $ABC$, $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD,BE$ are perpendicular to $CF$. $M$ is the midpoint of $AB$. If $ME=13$, $AD=15$, and $BE=25$, find $AC+CB$.

Source: 2012-2013 Winter Online Math Open