
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 \((1+\sqrt{2}+\sqrt{3})^{n}=p_{n}+q_{n} \sqrt{2}+r_{n} \sqrt{3}+s_{n} \sqrt{6}\) where \(n \geqslant 0\). Find:
a) \(\lim _{n \rightarrow \infty} \frac{p_{n}}{q_{n}}\);
b) \(\lim _{n \rightarrow \infty} \frac{p_{n}}{r_{n}}\);
c) \(\lim _{n \rightarrow \infty} \frac{p_{n}}{s_{n}}\).

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