
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
Let \( A \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + x - 110 < 0 \) and \( B \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + 10x - 96 < 0 \). Suppose that the set of integer solutions of the inequality \( x^{2} + ax + b < 0 \) is exactly the set of integers contained in \( A \cap B \). Find the maximum value of \( \lfloor |a - b| \rfloor \).

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