
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 $a \in \mathbb{R}$, the function $f(x) = 2x^3 - 3(a+1)x^2 + 6ax$  
(1) If $f(x)$ is monotonically increasing on $\mathbb{R}$, find the set of values for $a$;  
(2) If $|a| > 1$, find the minimum value $g(a)$ of $f(x)$ on the closed interval $[0, 2|a|]$.

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