Alice and Bob stand on opposite vertices of a regular octahedron. At the beginning of every minute,

- each chooses an adjacent vertex uniformly at random and moves towards it;
- each moves at a constant rate of 1 edge per minute;
- they will stop when they meet up.

The expected value of the number of minutes until they meet up is equal to \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers.

Find the value of \(p + q.\)

\(\)

**Note**: It is possible that they would meet at a vertex *or* at the midpoint of an edge.

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