
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
A unit square is divided by lines parallel to its sides into 9 equal parts, and the central part is removed. Each of the remaining 8 smaller squares is also divided by lines parallel to its sides into 9 equal parts, and the central part is removed. This process is then repeated for each remaining small square. After repeating this process \( n \) times, answer the following questions:
(1) How many squares with side length \( \frac{1}{3^{n}} \) are there?
(2) When \( n \) increases indefinitely, what is the limit of the sum of the areas of the squares that have been removed after \( n \) times?

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