
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 a positive integer  $n$ , let  $\tau(n)$  denote the number of positive divisors of  $n$  and  $\varphi(n)$  denote the number of positive integers not exceeding  $n$  that are relatively prime to  $n$ . Find all  $n$  for which one of the three numbers  $n,\tau(n), \varphi(n)$  is the arithmetic mean of the other two.

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