
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
Find all positive integer triples \((a, b, c)\) such that:
$$
\begin{array}{l}
a^{2}+b^{2}=n[a, b]+n^{2}, \\
b^{2}+c^{2}=n[b, c]+n^{2}, \\
c^{2}+a^{2}=n[c, a]+n^{2}
\end{array}
$$

for some positive integer \(n\).

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