
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  $\mathbb{Z}/n\mathbb{Z}$  denote the set of integers considered modulo  $n$  (hence  $\mathbb{Z}/n\mathbb{Z}$  has  $n$  elements). Find all positive integers  $n$  for which there exists a bijective function  $g: \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ , such that the 101 functions
\[g(x), \quad g(x) + x, \quad g(x) + 2x, \quad \dots, \quad g(x) + 100x\]
are all bijections on  $\mathbb{Z}/n\mathbb{Z}$ .

*Ashwin Sah and Yang Liu*

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