
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 a necessary and sufficient condition of  $a,b,n\in\mathbb{N^*}$  such that for  $S=\{a+bt\mid t=0,1,2,\cdots,n-1\}$ , there exists a one-to-one mapping  $f: S\to S$  such that for all  $x\in S$ ,  $\gcd(x,f(x))=1$ .

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