
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 $f(x) = 2x^3 - 3(k+1)x^2 + 1$ ($x \in \mathbb{R}$),  
(1) If the function has an extremum at $x = -1$, find the value of the real number $k$;  
(2) Determine the intervals of monotonicity for $f(x)$;  
(3) Find the minimum value of $f(x)$ on the interval $[0, 1]$.

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