
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
Determine all finite nonempty sets \( S \) of positive integers satisfying
\[ \frac{i+j}{(i, j)} \text{ is an element of } S \text{ for all } i, j \text{ in } S \]
where \((i, j)\) is the greatest common divisor of \(i\) and \(j\).

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