
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 $f(x) = 2x^3 - 3(a+1)x^2 + 6ax$, where $a$ is a positive number.  
$(1)$ Discuss the monotonicity of $f(x)$.  
$(2)$ If the maximum value of the function $y = f(x)$ on the closed interval $[0, a+1]$ is $f(a+1)$, find the range of values for $a$.

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