
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 \(f(x) = x^3 + ax^2 + bx + c\), whose all roots are non-negative real numbers, it holds that for all \(x \geq 0\),
\[ 
f(x) \geq \lambda(x - a)^3 
\]
and determine when the equality holds.

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