
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]


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Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a differentiable monotonic function such that \(f(0)=0\), and it satisfies the inequality:

\[ f[f(x)] \geq (\lambda + 1) f(x) - \lambda x, \]

find \( f^{\prime}(0) \).

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