
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
There is a sequence of curves \( P_{0}, P_{1}, P_{2}, \cdots \). It is known that the shape enclosed by \( P_{0} \) is an equilateral triangle with an area of 1. \( P_{k+1} \) is obtained from \( P_{k} \) by the following operation: each side of \( P_{k} \) is divided into three equal parts, an equilateral triangle is constructed outward on the middle segment of each side, and then the middle segment is removed \( (k=0,1,2, \cdots) \). Let \( S_{n} \) denote the area of the shape enclosed by the curve \( P_{n} \).
(1) Find the general term formula of the sequence \( \left\{S_{n}\right\} \);
(2) Find \( \lim S_{n} \).

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