
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
An ant starts at one vertex of an octahedron and moves along the edges according to a similar rule: at each vertex, the ant chooses one of the four available edges with equal probability, and all choices are independent. What is the probability that after six moves, the ant ends at the vertex exactly opposite to where it started?
A) $\frac{1}{64}$
B) $\frac{1}{128}$
C) $\frac{1}{256}$
D) $\frac{1}{512}$

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