HMMT 2025 G1

Equilateral triangles $\triangle ABC$ and $\triangle DEF$ are drawn such that points $B$ , $E$ , $F$ , and $C$ lie on a line in this order, and point $D$ lies inside triangle $\triangle ABC$ . If $BE = 14$ , $EF = 15$ , and $FC = 16$ , compute $AD$ .