
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
Expand the binomial $\left( \sqrt{x}+\dfrac{1}{2\sqrt[4]{x}} \right)^{n}$ and arrange the expansion in descending powers of $x$. If the coefficients of the first three terms form an arithmetic sequence, determine the number of terms in the expansion where the exponent of $x$ is an integer.

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