
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. Define $\varphi(n)$ as the number of positive integers that are less than or equal to $n$ and are coprime with $n$, and $\tau(n)$ as the number of positive divisors of $n$. Find all positive integers $n$ such that one of the numbers $n$, $\varphi(n)$, or $\tau(n)$ is the arithmetic average of the other two.

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