Lost in the Woods

You’re lost in the woods, and your only hope of reaching civilization is to find your way back to the only road.

• You know you are exactly 1000 meters from the infinitely long, straight road.
• You have perfect relative bearings, are able to track your position perfectly, and can travel along any line or curve.
• You don't know the direction of the road and cannot see it until you are upon it.

What is the shortest possible walking distance, to the nearest meter, that will ensure you reach the road?

Be warned, your initial thoughts may need to be refined to truly minimize the length of your path.