
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
Let n be a positive integer. Find all complex numbers  $x_{1}$ ,  $x_{2}$ , ...,  $x_{n}$  satisfying the following system of equations:

 $x_{1}+2x_{2}+...+nx_{n}=0$ ,
 $x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$ ,
...
 $x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$ .

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