
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
Given the sequence \(\left\{a_{n}\right\}\) satisfying \(a_{1}=1\) and \(a_{n+1}=\frac{a_{n}}{2}+2^{n-1} \cdot a_{n}^{2}\) for \(n \in \mathbf{N}^{*}\). Let \(S_{n}=\sum_{i=1}^{n} \frac{1}{2^{n} a_{n}+1}\). Find \(\lim _{n \rightarrow \infty} S_{n}\).

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