
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
Using a standard cube with vertices labeled from \( A \) to \( H \), eight ants simultaneously stand on each vertex of the cube. Each ant will move simultaneously and independently to an adjacent vertex shared by an edge with the vertex they initially occupy. The cube vertices are structured such that \( ABCD \) and \( EFGH \) form two parallel squares. What is the probability that no two ants meet at the same vertex?
A) \(\frac{1}{6561}\)
B) \(\frac{1}{729}\)
C) \(\frac{1}{2187}\)
D) \(\frac{1}{4096}\)
E) \(\frac{1}{1024}\)

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