
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 in the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[3]{x}})^n$, the coefficients of the first $3$ terms are $a_{1}$, $a_{2}$, $a_{3}$, and satisfy $2a_{2}=a_{1}+a_{3}$. Find:<br/>
$(Ⅰ)$ The sum of the binomial coefficients of all terms in the expansion;<br/>
$(Ⅱ)$ The term with the largest coefficient in the expansion;<br/>
$(Ⅲ)$ All rational terms in the expansion.

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