
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

Let \(m\) be a fixed even positive integer. Find all positive integers \(n\) for which there exists a bijection \(f\) from \(\{1, \ldots, n\}\) to itself such that for all \(x, y \in \{1, \ldots, n\}\) with \(m x - y\) divisible by \(n\), we also have
$$
(n+1) \mid f(x)^{m}-f(y).
$$

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