
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 real-coefficient quadratic equation \( x^{2} + (a+1)x + a+b+1 = 0 \) with two real roots \( x_{1} \) and \( x_{2} \), where \( 0 < x_{1} < 1 \) and \( x_{2} > 1 \), determine the range of \(\frac{b}{a}\).

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