
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 quadratic function $f(x)=x^{2}+2bx+c(b,c∈R)$ satisfies $f(1)=0$, and the two real roots of the equation $f(x)+x+b=0$ with respect to $x$ are respectively in the intervals $(-3,-2)$, $(0,1)$, the range of the real number $b$ is _____ .

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