An ant is walking along the edges of a polyhedron. It begins at one vertex and proceeds along an edge until it arrives at another vertex. It then randomly chooses an edge to proceed along (including back along the edge it came in on), etc. The ant's journey ends when it returns to the vertex at which it started.

For each Platonic solid (the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron), determine the expected length of the ant's trip.

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