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

Let's think step by step and output the final answer within \boxed{}.
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