
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 $a_{n}= \begin{cases} \frac {2^{n+1}}{2^{n}+1},1\leqslant n < 10000 \\ \frac {(n+1)^{2}}{n^{2}+1},n\geqslant 10000\end{cases}$, where $n\in\mathbb{N}^{*}$, find $\lim\limits_{n\to\infty}a_{n}=$ __.

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