
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
Given the function $f(x) = -\frac{1}{3}x^3 + ax^2 - bx$ $(a, b \in \mathbb{R})$.  
$(1)$ When $a = 1$, if for all $x > 0$ the inequality $f(x) \leq bx^2 + x$ holds, find the minimum value of the real number $b$.  
$(2)$ Let $b = -3a^2$ $(a > 0)$. If $x_1$ and $x_2$ are respectively the local minimum and local maximum points of the function $f(x)$,  
$①$ Calculate $f(x_1)$ and $f(x_2)$.  
$②$ When $\lambda \in (0, 1)$, find the range of $f\left(\frac{x_1 + \lambda x_2}{1 + \lambda}\right)$.

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