
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 $f(x) = 2x^3 + 3ax^2 + 1$ $(x \in \mathbb{R})$.

(Ⅰ) If $f(x)$ has an extreme value at $x = 1$, find the value of the real number $a$;  
(Ⅱ) Determine the intervals of monotonicity for $f(x)$;  
(Ⅲ) Find the minimum value of the function $f(x)$ on the closed interval $[0, 2]$.

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