Olympiad Geometry · Superchallenge problem

Town $A$ is at $(0, 0)$ and town $B$ is at $(12, 10)$. An east-west river has straight banks along $y = 2$ and $y = 4$, and a north-south river has banks along $x = 6$ and $x = 7$. A road from $A$ to $B$ crosses each river exactly once, on a bridge perpendicular to that river's banks; everywhere else the road may run in any direction. Find the length of the shortest possible road.