
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 $b, c \in \mathbb{R}$) satisfies $f(1) = 0$, and the two real roots of the equation $f(x) + x + b = 0$ are respectively 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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