
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 ${\left(2x+\frac{2}{{\sqrt{x}}}\right)^n}$, the sum of the binomial coefficients of the first three terms is $22$;  
$(1)$ Find the value of $n$;  
$(2)$ Find the term with the largest binomial coefficient in the expansion;  
$(3)$ Find the constant term in the expansion.

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