
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  $S$  be the set of positive integers  $n$  such that 
\[3\cdot
 \varphi (n)=n,\]
where  $\varphi (n)$  is the number of positive integers  $k\leq n$  such that  $\gcd (k, n)=1$ . Find
\[\sum_{n\in S} \, \frac{1}{n}.\]

[i]Proposed by Nathan Ramesh


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