
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
For each natural number  $n\geq 2$ , solve the following system of equations in the integers  $x_1, x_2, ..., x_n$ :  $$ (n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n $$  where  $$ S=x_1^2+x_2^2+\dots+x_n^2. $$  

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