Triangle $ABC$ has side lengths $AB=5$, $BC=12$, and $AC=13$. The points $D$, $E$, and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$ respectively. Let $X\ne E$ be the intersection of the circumcircles of $△BDE$ and $△CEF$. What is $XA+XB+XC$?