
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
The numbers \(a\) and \(b\) are such that each of the two quadratic polynomials \(x^2 + ax + b\) and \(x^2 + bx + a\) has two distinct roots, and the product of these polynomials has exactly three distinct roots. Find all possible values for the sum of these three roots.

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