
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 expansion of the binomial $({x+\frac{a}{{\sqrt{x}}}})^n$ where $n\in{N^*}$, in the expansion, ___, ___. Given the following conditions: ① the ratio of the binomial coefficients of the second term to the third term is $1:4$; ② the sum of all coefficients is $512$; ③ the $7$th term is a constant term. Choose two appropriate conditions from the above three conditions to fill in the blanks above, and complete the following questions.  
$(1)$ Find the value of the real number $a$ and the term with the largest binomial coefficient in the expansion;  
$(2)$ Find the constant term in the expansion of $({x\sqrt{x}-1}){({x+\frac{a}{{\sqrt{x}}}})^n}$.

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