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

(1) Find all rational terms in the expansion;

(2) Calculate the value of $n+9c_n^2+81c_n^3+\ldots+9^{n-1}c_n^n$.

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