Six towns math problem

I thought of this on my way to work and did the math in my head. When I get to a piece of paper I’ll see if I came up with the right answer.

Suppose there are six towns located at the points of a regular hexagon two miles on a side. Every town has a straight road to every other town, and there are no other roads.

People live and work along each of the roads (not just in the towns), and everyone bicycles to work along the shortest route available.

What is the longest distance anyone could possibly have to bicycle to get to work?

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