
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
Eight ants simultaneously stand on the eight vertices of a regular cube, each at a different vertex. Simultaneously and independently, each ant moves to one of the three adjacent vertices (not diagonal), each with equal probability. What is the probability that no two ants arrive at the same vertex?
A) $\frac{405}{6561}$
B) $\frac{405}{59049}$
C) $\frac{405}{43046721}$
D) $\frac{405}{4782969}$
E) $\frac{80}{59049}$

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