
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 graph of the function $f(x) = x^3 - 3ax^2 + 3a^2x - a^3$ ($a \in \mathbb{R}$) is symmetric about the point $(1, 0)$.

(1) Determine the analytical expression of $f(x)$;  
(2) Find the maximum and minimum values of the function $g(x) = f(x) - 2x^2$ on the interval $[-1, 1]$.

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