
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 binomial expression ($$\sqrt{x} + \sqrt[3]{x}$$)<sup>n</sup> (where $n \in \mathbb{N}^*$ and $n < 15$),  
(1) Find the sum of the coefficients of the terms in the expansion of the binomial expression;  
(2) If the binomial coefficients of the 9th, 10th, and 11th terms in the expansion form an arithmetic sequence, find the value of $n$;  
(3) Under the condition of (2), write out the rational terms in its expansion.

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