Rectangle $ABCD$ has $AB=7$ and $BC=6$. Point $E$ lies on $AB$ so that $EB=3$, point $G$ lies on $BC$ so that $CG=2$, and point $F$ lies on $CD$ so that $DF=4$. Segments $AG$ and $AC$ intersect $EF$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?