
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 $f\left(x\right)=x^{2}-2bx+3$, where $b\in R$.  
$(1)$ Find the solution set of the inequality $f\left(x\right) \lt 4-b^{2}$.  
$(2)$ When $x\in \left[-1,2\right]$, the function $y=f\left(x\right)$ has a minimum value of $1$. Find the maximum value of the function $y=f\left(x\right)$ when $x\in \left[-1,2\right]$.

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