You are driving along a perfectly straight road through the woods and decide the trees look like an inviting hike. After all, you have a GPS that could easily get you back to the road. So you get out and head off in a straight line perpendicular to the road, not paying any attention to your direction because, hey, GPS.

Unfortunately after traveling 1km your GPS crashes. It loses all of its map data as well as any previous journeys. In you panic, you even forgot which direction you were walking.

So here you are: 1 km away from the long straight road (the only one around for many km) in an unknown direction. You have a GPS that can still give your accurate position and path relative to your start.

What is the length of the shortest path (measured from here) that guarantees you will find the road?

Note: the trees are dense enough that you could be very close to the road and not see it.