
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
For an infinite geometric sequence $\{a_n\}$ ($n\in\mathbb{N}^*$) with the first term $a_1=1$ and common ratio $q=\frac{1}{3}$, find the limit of the sum of the first $n$ terms, $\lim_{n\to\infty}S_n$.

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