
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 $n \in \mathbf{N}^{*}$ such that the system of equations $\left\{\begin{array}{l}x_{1}+x_{2}+\cdots+x_{n}=9 \\ \frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}=1\end{array}\right.$ has positive solutions $x_{1}, x_{2}, \cdots, x_{n}$. For the obtained $n$, provide the corresponding solutions.

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