
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
From \(3n\) distinct elements \(a_{1}, a_{2}, \cdots, a_{n}, b_{1}, b_{2}, \cdots, b_{n}, c_{1}, c_{2}, \cdots, c_{n}\), how many permutations among the \((3n)!\) permutations satisfy the condition that for any natural number \(k\) (\(k=1, 2, \cdots, n\)), \(a_{k}\) is not adjacent to \(b_{k}\)?

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