
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
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule: At each vertex, the bug can choose to travel along any of the three edges emanating from that vertex, each with equal probability. What is the probability that after six moves, the bug ends up at a vertex opposite to the starting vertex?
A) $\frac{1}{81}$
B) $\frac{10}{243}$
C) $\frac{20}{243}$
D) $\frac{5}{243}$
E) $\frac{40}{2187}$

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