
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 a sequence $\{a_n\}$ with the first term $a_1 = 1$ and the sum of the first $n$ terms as $S_n$. It is known that the vectors $\overrightarrow{a} = (1, a_n)$ and $\overrightarrow{b} = (a_{n+1}, \frac{1}{2})$ are orthogonal, $\overrightarrow{a} \perp \overrightarrow{b}$. Find the limit $\lim_{n \to \infty} S_n$.

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