
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
What is the largest possible k such that \( p(x) \geq k(x - a)^3 \) for all non-negative \( x \) and all real polynomials \( p(x) = x^3 + ax^2 + bx + c \) which have real coefficients and real, non-negative roots? For which polynomials is equality achieved?

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