
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]


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A set \( S \) of positive integers is called magic if for any two distinct members of \( S \), \( i \) and \( j \),
\[ 
\frac{i+j}{\text{GCD}(i, j)}
\]
is also a member of \( S \). The \(\text{GCD}\), or greatest common divisor, of two positive integers is the largest integer that divides evenly into both of them; for example, \(\text{GCD}(36,80) = 4\).
Find and describe all finite magic sets.

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