
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
Solve the inequality \(\frac{\sqrt{\frac{x}{\gamma}+(\alpha+2)}-\frac{x}{\gamma}-\alpha}{x^{2}+a x+b} \geqslant 0\).

Indicate the number of integer roots of this inequality in the answer. If there are no integer roots or there are infinitely many roots, indicate the number 0 on the answer sheet.

Given:
\[
\alpha = 3, \gamma = 1, a = -15, b = 54.
\]

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