
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 expansion of $\left( \frac{1}{2}x^{2}- \frac{1}{\sqrt{x}} \right)^{n}$, where the 9th term is a constant term. Find:

(1) the value of $n$;

(2) the coefficient of $x^{5}$ in the expansion;

(3) the number of terms containing integral powers of $x$.

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