
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
Let \((1+\sqrt{2}+\sqrt{3})^{n}=q_{n}+r_{n} \sqrt{2}+s_{n} \sqrt{3}+t_{n} \sqrt{6}\), where \(q_{n}, r_{n}, s_{n}, t_{n}\) are natural numbers. Compute the limits \(\lim _{n \rightarrow \infty} \frac{r_{n}}{q_{n}}, \lim _{n \rightarrow \infty} \frac{s_{n}}{q_{n}}\), and \(\lim _{n \rightarrow \infty} \frac{t_{n}}{q_{n}}\).

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