HMMT 2025 U15

Right triangle $\triangle DEF$ with $\angle D = 90^{\circ}$ and $\angle F = 30^{\circ}$ is inscribed in equilateral triangle $\triangle ABC$ such that $D$ , $E$ , and $F$ lie on segments $\overline{BC}$ , $\overline{CA}$ , and $\overline{AB}$ , respectively. Given that $BD = 7$ and $DC = 4$ , compute $DE$ .