
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
The expansion of the binomial \(\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right)^{n}\), arranged in descending order of powers of \(x\), has its first three coefficients forming an arithmetic sequence. Determine the number of terms in the expansion with integer exponents of \(x\).

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