
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 geometric sequence \(\left\{a_{n}\right\}\) with \(a_{1}>0\) and \(S_{n}=80\), the largest term among the first \(n\) terms is 54, and \(S_{2n} = 6560\). Find the general term \(a_{n}\) of this sequence and determine the limit \(\lim _{n \rightarrow \infty} \frac{a_{n}}{S_{n}}\).

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