
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 \( x_n = (1 + \sqrt{2} + \sqrt{3})^n \) for \( n = 1, 2, 3, \ldots \), and expressing \( x_n \) in the form \( x_n = q_n + r_n \sqrt{2} + s_n \sqrt{3} + t_n \sqrt{6} \) where \( q_n, r_n, s_n \), and \( t_n \) are integers, find the following limits:
$$
\lim_{n \rightarrow \infty} \frac{r_n}{q_n}, \lim_{n \rightarrow \infty} \frac{s_n}{q_n}, \lim_{n \rightarrow \infty} \frac{t_n}{q_n}.
$$

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