
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 positive solutions for the system for all natural \( n > 1 \):

$$
\left\{\begin{array}{l}
x_{1}+2 x_{2}+\cdots+n x_{n}=3 \\
\frac{1}{x_{1}}+\frac{1}{2 x_{2}}+\cdots+\frac{1}{n x_{n}}=3
\end{array}\right.
$$

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