
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 natural values of  $k$  for which the system  $$ \begin{cases}x_1+x_2+\ldots+x_k=9\frac1{x_1}+\frac1{x_2}+\ldots+\frac1{x_k}=1\end{cases} $$  has solutions in positive numbers. Find these solutions.

*I. Dimovski*

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