
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 natural numbers \( n \) such that there exists a permutation \((p_1, p_2, \ldots, p_n)\) of \((1, 2, \ldots, n)\) that satisfies: the sets \(\{p_i + i \mid 1 \leq i \leq n\}\) and \(\{p_i - i \mid 1 \leq i \leq n\}\) are both complete residue systems modulo \( n \).

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