
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 $a_n = \begin{cases} \frac{2^{n+1}}{2^n+1}, & 1 \leq n < 10000 \\ \frac{(n+1)^2}{n^2+1}, & n \geq 10000 \end{cases}$, where $n \in \mathbb{N}^*$, then $\lim_{n \to \infty} a_n = \_\_\_\_\_\_$.

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