
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 $f\left(x\right)=\frac{1}{3}{x}^{3}-a{x}^{2}+\left({a}^{2}-1\right)x+b\left(a,b∈R\right)$

(1) If $x=1$ is the extreme point of $f(x)$, find the value of $a$;

(2) If the equation of 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 on the interval $(-1,1)$, find the range of values for $a$.

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