Olympiad Geometry · Superchallenge problem

In $\triangle ABC$, $m\angle A = 60^\circ$. The bisector of $\angle B$ meets $\overline{AC}$ at $E$, and the bisector of $\angle C$ meets $\overline{AB}$ at $F$. If $BF = 5$ and $CE = 3$, find $BC$.