
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 $f\left(x\right)=x^{2}+bx+c\left(b,c\in R\right)$, where $f\left(x\right)-x=0$ has two distinct roots $x_{1}$ and $x_{2}$, and $(x_{1}-1)(x_{2}-1)=5$.  
$(1)$ If $x_{1}=x_{2}+4$, find the value of $b$.  
$(2)$ Find the range of values for $x_1^2+x_2^2$.

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