
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
Determine all sequences of integers  $a_1, a_2,. . .,$  such that:
(i)  $1 \le  a_i \le n$  for all  $1 \le i \le n$ .
(ii)  $| a_i - a_j| = | i - j |$  for any  $1 \le i, j \le n$ 

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