
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
Find all odd positive integers  $n>1$  such that there is a permutation  $a_1, a_2, a_3, \ldots, a_n$  of the numbers  $1, 2,3, \ldots, n$  where  $n$  divides one of the numbers  $a_k^2 - a_{k+1} - 1$  and  $a_k^2 - a_{k+1} + 1$  for each  $k$ ,  $1 \leq k \leq n$  (we assume  $a_{n+1}=a_1$ ).

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