
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 that the coefficients of the first three terms in the expansion of ${(1+\frac{1}{2}x)}^{n}$ form an arithmetic sequence.<br/>$(1)$ Find the value of $n$;<br/>$(2)$ Find the coefficient of $x^{2}$ in the expansion of ${(1+\frac{1}{2}x)}^{n}{(1-x)}^{2}$.

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