In rectangle $ABCD$, we have $AB=8$ and $BC=9$. $H$ is on $BC$ with $BH=6$, $E$ is on $AD$ with $DE=4$. Line $EC$ intersects line $AH$ at $G$, and $F$ is on line $AB$ with $GF\perp AF$. Find the length of $GF$. Refer to the diagram provided with this problem, which illustrates the general setup.