
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 ${(\sqrt{x}+\sqrt[3]{x})^{n}}$ where $n < 15$, $n \in \mathbb{N}^*$, the binomial coefficients of the 9th, 10th, and 11th terms form an arithmetic sequence.

(1) Find the value of $n$;
(2) Write out all the rational terms in its expansion.

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