Two Bugs on an Icosahedron

Two bugs start on adjacent vertices of an icosahedron. Every "move" they each randomly walk along one of the five edges available to them. What is the expected number of moves for them to meet (on either an edge or a vertex)?

Assume that if they meet on an edge, that counts as a full move, even though they have each only traversed half of the edge.

If the answer is \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is \(a+b\)?

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