
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 \(a, b, c\) be positive integers. All the roots of each of the quadratics
\[ a x^{2}+b x+c, \quad a x^{2}+b x-c, \quad a x^{2}-b x+c, \quad a x^{2}-b x-c \]
are integers. Over all triples \((a, b, c)\), find the triple with the third smallest value of \(a+b+c\).

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