
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 that the function $f(x) = ax^3 + bx^2 - 3x$ ($a, b \in \mathbb{R}$) has a local maximum of 2 at $x = -1$,  
(1) find the analytic expression for $f(x)$;  
(2) if for any two values of the independent variable $x_1, x_2$ in the interval $[-2, 2]$, it holds that $|f(x_1) - f(x_2)| \leq c$, find the minimum value of the real number $c$.

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