
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
1. \(\lim_{x \rightarrow \infty} \frac{3 x^{2} - 1}{5 x^{2} + 2 x}\)

2. \(\lim_{n \rightarrow -\infty} \frac{n}{\sqrt{n^{2} + 1}}\)

3. \(\lim_{n \rightarrow +\infty} \frac{1 + 7^{n + 2}}{3 - 7^{n}}\)

4. \(\lim_{n \rightarrow +\infty} \frac{2 + 4 + 6 + \ldots + 2n}{1 + 3 + 5 + \ldots + (2n + 1)}\)

5. \(\lim_{x \rightarrow \frac{\pi}{4}} \frac{\operatorname{tg} 2x}{\operatorname{ctg}\left(\frac{\pi}{4} - x\right)}\)

6. \(\lim_{n \rightarrow +\infty} \frac{n^{3}}{n^{2} + 2^{2} + 3^{2} + \ldots + n^{2}}\)

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