
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 quadratic polynomials $ax^2 + bx + c = 0$ with distinct non-zero real coefficients a, b, c, determine how many such polynomials exist such that one of the coefficients equals the product of the roots, and the other coefficients are the roots themselves.

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