
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 $f(x)=\frac{1}{3}x^{3}-ax^{2}+(a^{2}-1)x+b$ where $a, b \in \mathbb{R}$, and the tangent line at the point $(1, f(1))$ is $x+y-3=0$.  
$(1)$ Find the values of $a$ and $b$;  
$(2)$ Determine the intervals of monotonicity and extreme values of the function $f(x)$;  
$(3)$ Find the maximum value of the function $f(x)$ on the interval $[-2, 5]$.

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