Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be a point on $\overline{AB}$ such that $AD=\frac{2}{5}AB$, and let $E$ be a point on $\overline{AC}$ such that $AE=\frac{2}{5}AC$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?