KöMaL 2025 C1879

A convex $n$-gon is given on the table, the sides of which have length 2. We place a disk with unit radius on each vertex of the polygon such that the adjacent disks are tangent to each other. We roll another circular token with unit radius around this arrangement such that it returns to its initial position. Determine, as a function of $n$, the total angle through which this disk rotates during the rolling process.