
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 the quadratic polynomial \( ax^2 + bx + c = 0 \), with real coefficients, roots \( r \) and \( s \), and \( a \neq 0 \), determine the number of such polynomials where the set of roots equals the set of coefficients and at least one of the coefficients is negative.

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