
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 $2C_n^1+2^2C_n^2+2^3C_n^3+\cdots+2^nC_n^n=728$.  
$(1)$ Find the value of $n$;  
$(2)$ Find the constant term in the binomial expansion of $(x+\frac{1}{2\sqrt{x}})^n$.

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