
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 the system of equations:
\[
\begin{cases}
x + y + z = 0 \\
c x + a y + b z = 0 \\
(x + b)^2 + (y + c)^2 + (z + a)^2 = a^2 + b^2 + c^2
\end{cases}
\]
where \(a \neq b \neq c\). 

Find the values of \(x\), \(y\), and \(z\).

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