
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
Given a permutation \(\left(a_{0}, a_{1}, \ldots, a_{n}\right)\) of the sequence \(0,1, \ldots, n\). A transposition of \(a_{i}\) with \(a_{j}\) is called legal if \(a_{i}=0\) for \(i>0\), and \(a_{i-1}+1=a_{j}\). The permutation \(\left(a_{0}, a_{1}, \ldots, a_{n}\right)\) is called regular if after a number of legal transpositions it becomes \((1,2, \ldots, n, 0)\). For which numbers \(n\) is the permutation \((1, n, n-1, \ldots, 3,2,0)\) regular?

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