CF696F ...Dary!

Barney has finally found the one, a beautiful young lady named Lyanna. The problem is, Lyanna and Barney are trapped in Lord Loss' castle. This castle has shape of a convex polygon of $ n $ points. Like most of castles in Demonata worlds, this castle has no ceiling. Barney and Lyanna have an escape plan, but it requires some geometry knowledge, so they asked for your help. Barney knows that demons are organized and move in lines. He and Lyanna want to wait for the appropriate time so they need to watch for the demons. Each of them wants to stay in a point inside the castle (possibly on edges or corners), also they may stay in the same position. They both want to pick a real number $ r $ and watch all points in the circles with radius $ r $ around each of them (these two circles may overlap). We say that Barney and Lyanna are watching carefully if and only if for every edge of the polygon, at least one of them can see at least one point on the line this edge lies on, thus such point may not be on the edge but it should be on edge's line. Formally, each edge line should have at least one common point with at least one of two circles. The greater $ r $ is, the more energy and focus they need. So they asked you to tell them the minimum value of $ r $ such that they can watch carefully.

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In the first example guys can stay in opposite corners of the castle.