
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
For each pair of values \(n, k \in \mathbf{N}\), find all sets of non-negative numbers \(x_{1}, x_{2}, \ldots, x_{n}\) that satisfy the system

\[ \left\{
\begin{array}{l}
x_{1}^{k}+x_{2}^{k}+\ldots+x_{n}^{k}=1 \\
\left(1+x_{1}\right)\left(1+x_{2}\right) \ldots\left(1+x_{n}\right)=2
\end{array}
\right. \]

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