
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
Let \( f(x, y, z) \) be a polynomial in \( x, y, z \) and a 4th degree polynomial in \( x \), satisfying the conditions:
$$
\left\{\begin{array}{l}
f\left(x, z^{2}, y\right) + f\left(x, y^{2}, z\right) = 0, \\
f\left(z^{3}, y, x\right) + f\left(x^{3}, y, z\right) = 0.
\end{array}\right.
$$

Find such a polynomial \( f(x, y, z) \).

### Solution


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