
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 \( n \) for which there exists a permutation \(\left(p_{1}, p_{2}, \ldots, p_{n}\right)\) of \(\{1, 2, \ldots, n\}\) such that the sets \(\left\{p_{i}+i \mid 1 \leq i \leq n\right\}\) and \(\left\{p_{i}-i \mid 1 \leq i \leq n\right\}\) are complete residue systems modulo \( n \).

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