
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 $n\in \mathbb{N}^{*}$, consider the binomial ${(\sqrt{x}+\frac{1}{2\cdot\sqrt[4]{x}})^n}$.  
$(1)$ If the binomial coefficient of the $4$th term and the $8$th term in the expansion are equal, find the coefficient of $x^{2}$ in the expansion.  
$(2)$ If the coefficients of the first three terms in the expansion form an arithmetic sequence, find the term with the largest coefficient in the expansion.

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