
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
Given \( n \) as a positive integer and \( x_{0} \) as an integer, for each prime factor \( p \) of \(\Phi_{n}(x_{0})\), either \( n \) divides \( p-1 \) or \( p \) divides \( n \).

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