A Neurotic Bug

Probability Level 4

Let A,B,CA, B, C and DD be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex AA, observes the following rule:

At each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely, and crawls along that edge to the vertex at its opposite end.

Suddenly the bug remembers that it has left its antenna at AA.

Find the probability that the bug is at vertex AA when it has crawled exactly 7 meters.

If the probability is the form of pq \dfrac pq, where pp and qq are coprime positive integers, find p+qp+q.

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