
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 $\mathbf{Z} / n \mathbf{Z}$ denote the set of integers modulo $n$ (which has $n$ elements). Find all positive integers $n$ for which there exists a bijection $g: \mathbf{Z} / n \mathbf{Z} \rightarrow \mathbf{Z} / n \mathbf{Z}$ such that the 101 functions
$$
g(x), g(x) + x, g(x) + 2x, \cdots, g(x) + 100x
$$
are all bijections on $\mathbf{Z} / n \mathbf{Z}$.

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