
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
Let \( g_{n} \) denote the number of permutations of \( 2n \) distinct elements \( a_{1}, a_{2}, \cdots, a_{n}, b_{1}, b_{2}, \cdots, b_{n} \), such that \( a_{k} \) and \( b_{k} \) \((k = 1, 2, \cdots, n)\) are not adjacent for any \( k \). Find the formula for calculating \( g_{n} \).

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