A problem by Sam Thompson

Level pending

A bug is currently standing on vertex \(A\) of octahedron \(ABCDEF\), where \(F\) and \(A\) are the apexes of square pyramids \(FBCDE\) and \(ABCDE\), respectively. A move is counted when the bug travels to an adjacent vertex of the octahedron. Assuming that this is the only way the bug can travel the octahedron, and the bug is not biased towards any point, if \(\frac{m}{n}\) is the probability that the bug will be standing on vertex \(F\) after 9 moves, where \(m\) and \(n\) are coprime positive integers, then what is \(m+n\)?

Image Credit: Wikipedia

This problem was adapted from SMaC Contest 1 #6.

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