
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 geometric sequence \(\{a_{n}\}\) where each term is positive, the first term is 1 and the common ratio is \(x\). The sum of the first \(n\) terms is \(S_{n}\). Define \(f(x) = \lim _{n \rightarrow \infty} \frac{2 S_{n}}{S_{n+1}}\).

1. Find the analytical expression of \(f(x)\).
2. Plot the graph of the function \(y = f(x)\).

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