
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
The function $f(x) = x^3 + 3ax^2 + 3bx + c$ has an extremum at $x = 2$, and its graph has a tangent at $x = 1$ that is parallel to the line $3x + y + 2 = 0$.

(1) Find the values of $a$ and $b$; (2) Find the difference between the maximum and minimum values of the function.

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