
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
Determine all triplets of real numbers \((x, y, z)\) satisfying the system of equations
$$
\begin{aligned}
x^{2} y + y^{2} z & = 1040, \\
x^{2} z + z^{2} y & = 260, \\
(x - y)(y - z)(z - x) & = -540.
\end{aligned}
$$

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