
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 \(\left\{a_{n}\right\}\) with the first term 2 and common ratio \(\frac{1}{2}\), and the sum of the first \(n\) terms is \(S_{n}\):
1. Express \(S_{n+1}\) in terms of \(S_{n}\).
2. Do there exist natural numbers \(c\) and \(k\) such that \(\frac{S_{k+1}-c}{S_{k}-c} > 2\)?

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