
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 the following conditions: ① If the sum of the coefficients of the first three terms in the expansion is $16$; ② If the ratio of the coefficient of the third term from the end to the coefficient of the second term from the end is $4:1$. Choose one of them and solve the following problem.<br/>Given $({x+\frac{{\sqrt{x}}}{2}})^n$ where $n\in{N^*}$, _______.<br/>$(1)$ Find the term with the largest binomial coefficient in the expansion;<br/>$(2)$ Find all rational terms in the expansion.

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