
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 non-negative real roots, the inequality $f(x) \geqslant \lambda(x - a)^3$ holds for all $x \geq 0$. Also, determine when equality holds in this expression.

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