
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 $f(x) = x^{2} - 2ax + b$ has a local extremum of $2$ at $x=1$.  
$(1)$ Find the extreme values of the function $f(x) = x^{2} - 2ax + b$ on the closed interval $[0,3]$.  
$(2)$ Find the area $S$ enclosed by the curves $y=x^{2}-2ax+b$ and $y=x+3$.

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