HMMT 2018 G7

Triangle $A B C$ has sidelengths $A B=14, A C=13$, and $B C=15$. Point $D$ is chosen in the interior of $\overline{A B}$ and point $E$ is selected uniformly at random from $\overline{A D}$. Point $F$ is then defined to be the intersection point of the perpendicular to $\overline{A B}$ at $E$ and the union of segments $\overline{A C}$ and $\overline{B C}$. Suppose that $D$ is chosen such that the expected value of the length of $\overline{E F}$ is maximized. Find $A D$.