
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 point of $f(x)$, find the value of $a$;  
$(2)$ If the tangent line of the graph of $y=f(x)$ at the point $(1,f(1))$ is $x+y-3=0$, find the maximum and minimum values of $f(x)$ on the interval $[-2,4]$.

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