
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
Let $a_n = \begin{cases} 
2^{n-1}, & 1 \leq n \leq 2, n \in \mathbb{N} \\
\frac{1}{3^n}, & n \geq 3, n \in \mathbb{N}
\end{cases}$ be a sequence. Find the limit of the sum of the first $n$ terms of the sequence, $S_n$, as $n$ approaches infinity, i.e., $\lim_{n \to \infty} S_n = \_\_\_\_\_$.

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