
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
Find the maximum real number \(\lambda\) such that for the real-coefficient polynomial
$$
f(x) = x^3 + ax^2 + bx + c
$$
with all roots being non-negative real numbers, the inequality
$$
f(x) \geqslant \lambda(x - a)^3 \quad \text{for all} \; x \geqslant 0
$$
holds. Also, determine when equality holds in this inequality.

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