HMMT 2006 U6

Suppose $A B C$ is a triangle such that $A B=13, B C=15$, and $C A=14$. Say $D$ is the midpoint of $\overline{B C}, E$ is the midpoint of $\overline{A D}, F$ is the midpoint of $\overline{B E}$, and $G$ is the midpoint of $\overline{D F}$. Compute the area of triangle $E F G$.