
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
Given a positive integer  $n \geq 2$ . Solve the following system of equations: $
\begin{cases}
\ x_1|x_1| &= x_2|x_2| + (x_1-1)|x_1-1| 
\ x_2|x_2| &= x_3|x_3| + (x_2-1)|x_2-1| 
&\dots 
\ x_n|x_n| &= x_1|x_1| + (x_n-1)|x_n-1|. 
\end{cases}
$ 

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