
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 $f(x)=x^{2}+2bx+c$ where $b$ and $c$ are real numbers.  
$(Ⅰ)$ If the zeros of the function $f(x)$ are $-1$ and $1$, find the values of the real numbers $b$ and $c$.  
$(Ⅱ)$ Given $c=b^{2}+2b+3$, let $x_{1}$ and $x_{2}$ be the two roots of the equation $f(x)=0$, and $(x_{1}+1)(x_{2}+1)=8$, find the value of the real number $b$.  
$(Ⅲ)$ If $f(x)$ satisfies $f(1)=0$, and the two real roots of the equation $f(x)+x+b=0$ are in the intervals $(-3,-2)$ and $(0,1)$, find the range of values for the real number $b$.

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