
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the vertices of a regular cube each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three connecting edges, each edge being equally likely to be chosen, and crawls along that edge to reach the vertex at the other end. Let $p = \frac n{19683}$ be the probability that the bug is at vertex $A$ when it has crawled exactly 9 meters. Find the value of $n$.

Present the answer in LaTex format: \boxed{Your answer}
user