
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 integers \( x \), \( y \), and \( z \) satisfy the following system of equations:
\[ \left\{\begin{array}{l}
|x+y|+|y+z|+|z+x|=4, \\
|x-y|+|y-z|+|z-x|=2.
\end{array}\right. \]
Find the value of \( x^{2} + y^{2} + z^{2} \).

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