
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
Given positive integers \(a, b, c\) satisfying \(a<b<c\), find all triplets \((a, b, c)\) that satisfy the equation:
$$
\frac{(a, b)+(b, c)+(c, a)}{a+b+c} = \frac{1}{2}
$$
where \((x, y)\) denotes the greatest common divisor of the positive integers \(x\) and \(y\).

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