Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Point $D$ is on $\overline{AC}$ such that $AD=\frac{1}{3}AC$. Point $G$ is on $\overline{AE}$ such that $AG=\frac{1}{3}AE$. If $AB=9$, what is the perimeter of figure $ABCDEFG$?