
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 $f(x) = \frac{1}{3}x^3 - ax^2 + (a^2 - 1)x + b$ ($a, b \in \mathbb{R}$).  
(1) If $x=1$ is an extremum of $f(x)$, find the value of $a$;  
(2) If the tangent line to the graph of $y=f(x)$ at the point $(1, f(1))$ is $x+y-3=0$, find the maximum value of $f(x)$ on the interval $[-2, 4]$;  
(3) When $a \neq 0$, if $f(x)$ is not monotonic in the interval $(-1, 1)$, find the range of values for $a$.

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