
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 \(\varphi(n)\) denote the number of positive integers less than or equal to \(n\) which are relatively prime to \(n\). Let \(S\) be the set of positive integers \(n\) such that \(\frac{2n}{\varphi(n)}\) is an integer. Compute the sum
\[ \sum_{n \in S} \frac{1}{n} . \]

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