
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
Determine the number of quadratic polynomials $ax^2+bx+c$ with real coefficients for which the set of roots matches the set of coefficients, and additionally, one of the coefficients is the arithmetic mean of the other two.

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