
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 ( $a_0, a_1, \ldots, a_n$ ) of the sequence  $0, 1,\ldots, n$ . A transportation 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 ( $a_0, a_1, \ldots, a_n$ ) is called regular if after a number of legal transportations 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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