
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 the necessary and sufficient conditions for positive integers \(a\), \(b\), and \(n\) (where \(n \geq 2\)) such that there exists a bijective function \(f: S \rightarrow S\) from the set \(S=\{a+bt \mid t = 0, 1, \ldots, n-1\}\) to itself, satisfying: for any \(x \in S\), \(x\) and \(f(x)\) are coprime.

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