
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
If the general term of the sequence $\{a_n\}$ is given by
$$
a_n = \begin{cases} 
2^{n+1} & (1\leq n \leq 2) \\
\frac{1}{3^n} & (n\geq 3)
\end{cases},
$$
and the sum of the first $n$ terms is $S_n$, then the value of $\lim_{n\to\infty} S_n$ is ______.

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