
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 solutions of the following system of \( n \) equations in \( n \) variables:
$$
\begin{array}{c}
x_{1}\left|x_{1}\right|-\left(x_{1}-a\right)\left|x_{1}-a\right|=x_{2}\left|x_{2}\right|, \\
x_{2}\left|x_{2}\right|-\left(x_{2}-a\right)\left|x_{2}-a\right|=x_{3}\left|x_{3}\right|, \\
\cdots \\
x_{n}\left|x_{n}\right|-\left(x_{m}-a\right)\left|x_{m}-a\right|=x_{1}\left|x_{1}\right|,
\end{array}
$$
where \( a \) is a given number.

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