
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 sets of three real numbers \((x, y, z)\) that satisfy the following system of equations:

\[
\begin{cases}
x^{2}-yz = |y-z| + 1, \\
y^{2}-zx = |z-x| + 1, \\
z^{2}-xy = |x-y| + 1.
\end{cases}
\]

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