
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
Let the function $f(x) = -\frac{1}{3}x^3 + 2ax^2 - 3a^2x + b$ (where constants $a$, $b$ satisfy $0 < a < 1$, $b \in \mathbb{R}$).  
$(1)$ Find the intervals of monotonicity and the extremum of the function $f(x)$.  
$(2)$ If for any $x \in [a+1, a+2]$, the inequality $|f'(x)| \leqslant a$ always holds, find the range of $a$.

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