
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
Determine all natural numbers \( n \) so that for any real numbers \( x_{1}, \ldots, x_{n} \), the following inequality holds:

$$
\left(\frac{x_{1}^{n}+\ldots+x_{n}^{n}}{n}-x_{1} \cdot \ldots \cdot x_{n}\right)\left(x_{1}+\ldots+x_{n}\right) \geq 0
$$

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