
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 real numbers \( x, y, z \) such that

\[
\begin{aligned}
x^{2} y + y^{2} z + z^{2} &= 0, \\
z^{3} + z^{2} y + z y^{3} + x^{2} y &= \frac{1}{4}\left(x^{4} + y^{4}\right).
\end{aligned}
\]

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