
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
As shown in the diagram, there is a sequence of curves \(P_{0}, P_{1}, P_{2}, \cdots\). It is given that \(P_{0}\) is an equilateral triangle with an area of 1. Each \(P_{k+1}\) is obtained from \(P_{k}\) by performing the following operations: each side of \(P_{k}\) is divided into three equal parts, an equilateral triangle is constructed outwards on the middle segment of each side, and the middle segments are then removed (\(k=0,1,2, \cdots\)). Let \(S_{n}\) denote the area enclosed by the curve \(P_{n}\).
1. Find a general formula for the sequence \(\{S_{n}\}\).
2. Evaluate \(\lim _{n \rightarrow \infty} S_{n}\).

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