
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
Assume the function $f(x)=2x^3-3(a+1)x^2+6ax+8$ ($a \in \mathbb{R}$) attains an extreme value at $x=3$.

1. Determine the value of the constant $a$.
2. Find the intervals of monotonicity of $f(x)$ over $\mathbb{R}$.
3. Calculate the maximum and minimum values of $f(x)$ on the interval $[-4, 4]$.

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