
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 an arithmetic sequence $\{a_n\}$ with the first term $a$ ($a \in \mathbb{R}, a \neq 0$). Let the sum of the first $n$ terms of the sequence be $S_n$, and for any positive integer $n$, it holds that $$\frac {a_{2n}}{a_{n}}= \frac {4n-1}{2n-1}$$.
(1) Find the general formula for $\{a_n\}$ and $S_n$;
(2) Does there exist positive integers $n$ and $k$ such that $S_n$, $S_{n+1}$, and $S_{n+k}$ form a geometric sequence? If so, find the values of $n$ and $k$; if not, explain why.

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