
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 a quadratic function $f\left(x\right)=x^{2}+2bx+c\left(b,c\in R\right)$.<br/>$(1)$ If the zeros of the function $f\left(x\right)$ are $-1$ and $1$, find the values of real numbers $b$ and $c$;<br/>$(2)$ Given $c=b^{2}+2b+3$, let $x_{1}$ and $x_{2}$ be the two roots of the equation $f\left(x\right)=0$ in terms of $x$, and $(x_{1}+1)(x_{2}+1)=8$, find the value of the real number $b$.

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