
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
In the sequence $\left\{a_{n}\right\}$, where $a_{1}=1$, the sum of the first $n$ terms is $S_{n}$. For any integer $n \geqslant 2$, the sequence $S_{n-1}-8, a_{n}, 8 S_{n}-4$ is an arithmetic sequence. Find $\lim _{n \rightarrow \infty}\left(a_{n}+S_{n}\right)$.

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