
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 $n\in\mathbb{N}^*$, the area of the circle $C_n: \left(x - \frac{1}{n}\right)^2 + (y - 1)^2 = \frac{4^{n+1}-1}{4^n+1}$ is denoted as $S_n$. Calculate $\lim_{n \to +\infty} S_n$.

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