
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 \( n \) be a positive integer, \( \varphi(n) \) denote the number of positive integers less than or equal to \( n \) that are coprime with \( n \), and \( \tau(n) \) denote the number of positive divisors of \( n \). Find all positive integers \( n \) such that one of \( n \), \( \varphi(n) \), or \( \tau(n) \) is the arithmetic mean of the other two.

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