
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 \( a, b \), and \( c \) be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: \( a x^{2}+b x+c \), \( b x^{2}+c x+a \), and \( c x^{2}+a x+b \).

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