
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 sequence $\left\{a_{n}\right\}$ where each term is a non-zero real number, and for any positive integer $n$, the following holds:
\[
\left(a_{1}+a_{2}+\cdots+a_{n}\right)^{2}=a_{1}^{3}+a_{2}^{3}+\cdots+a_{n}^{3}.
\]

(1) When $n=3$, find all sequences satisfying the conditions with terms $a_{1}, a_{2}, a_{3}$.

(2) Does there exist an infinite sequence $\left\{a_{n}\right\}$ satisfying the conditions such that $a_{2013}=-2012$? If it exists, find the general term formula for such an infinite sequence; if it does not exist, provide a reason.

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