In rectangle $ABCD$, we have $AB=12$ and $BC=10$. $H$ is on $BC$ with $BH=8$, $E$ is on $AD$ with $DE=5$. Line $EC$ intersects line $AH$ at $G$, and $F$ is on line $AD$ with $GF\perp AF$. Find the length of $GF$. Refer to the diagram provided with this problem, which illustrates the general setup.