
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 in the expansion of $\left( \sqrt{x}- \frac{2}{ \sqrt[3]{x}}\right)^{n}$, the ratio of the coefficient of the 5th term to the coefficient of the 3rd term is $56:3$.

$(1)$ Find all the rational terms in the expansion;

$(2)$ Find the term with the largest absolute value of the coefficient in the expansion;

$(3)$ Find the value of $n+9C_{n}^{2}+81C_{n}^{3}+…+{9}^{n-1}C_{n}^{n}$.

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