
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 $(0 < a < 1)$.

$(1)$ Find the intervals of monotonicity for the function $f(x)$, and determine the maximum and minimum values of $f(x)$;

$(2)$ When $x \in [a+1,a+2]$, and the inequality $|f′(x)| \leqslant a$ holds, find the range of values for $a$.

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