
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
Let \( n \) be a given positive integer. Find all solutions to the following system of equations:

\[
x_{1}^{2 n} + \left(x_{1} x_{2}\right)^{2 n} + \ldots + \left(x_{1} x_{2} \ldots x_{n}\right)^{2 n} = 1
\]

\[
x_{1}^{2 n+1} + \left(x_{1} x_{2}\right)^{2 n+1} + \ldots + \left(x_{1} x_{2} \ldots x_{n}\right)^{2 n+1} = 1
\]

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