road path problem

Suppose there is a straight road, infinitely long at both ends, located 1 unit from your starting location. Find the most efficient path to reach the road, and the worst-case total length of this path.

The trivial but wrong way is to go for 1 unit in some direction, then trace the circumference of a unit-radius circle. The road will surely be found this way, but the path length is \(1+2\pi\), which can be improved upon.
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