
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
It is known that $S_n$ is the sum of the first $n$ terms of the geometric sequence $\{a_n\}$. $S_4$, $S_2$, and $S_3$ form an arithmetic sequence, and $a_2 + a_3 + a_4 = -18$.

(1) Find the general formula for the sequence $\{a_n\}$.

(2) Determine whether there exists a positive integer $n$ such that $S_n \geq 2013$. If it exists, find the set of all such $n$; if not, explain why.

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