
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
Find all positive integers \( n \) such that there exist nonzero integers \( x_1, x_2, \ldots, x_n, y \) satisfying
$$
\left\{
\begin{array}{l}
x_1 + x_2 + \cdots + x_n = 0, \\
x_1^2 + x_2^2 + \cdots + x_n^2 = n y^2.
\end{array}
\right.
$$

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