
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 integers \( n \geq 2 \) for which the system:

\[
\left\{\begin{aligned}
x_{1}^{2}+x_{2}^{2}+50 & =16 x_{1}+12 x_{2} \\
x_{2}^{2}+x_{3}^{2}+50 & =16 x_{2}+12 x_{3} \\
& \vdots \\
x_{n-1}^{2}+x_{n}^{2}+50 & =16 x_{n-1}+12 x_{n} \\
x_{n}^{2}+x_{1}^{2}+50 & =16 x_{n}+12 x_{1}
\end{aligned}\right.
\]

has at least one solution in integer numbers.

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