
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 values of \( n \in \mathbf{N} \) for which there exists a set of positive numbers \( x_{1}, \ldots, x_{n} \) that satisfies the system:

\[
\left\{
\begin{array}{l}
x_{1} + x_{2} + \ldots + x_{n} = 9 \\
\frac{1}{x_{1}} + \frac{1}{x_{2}} + \ldots + \frac{1}{x_{n}} = 1
\end{array}
\right.
\]

Indicate all such sets for each found value of \( n \).

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