A convex hexagon $ABCDEF$ has side lengths $AB=5$, $BC=2$, $CD=4$, $DE=3$, $EF=6$, and $FA=2$. The hexagon is originally positioned in the plane with vertex $A$ at the origin and vertex $B$ on the positive $x$-axis. The hexagon is then rolled clockwise to the right along the $x$-axis. Which side will touch the point $x=1000$ on the $x$-axis?