
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 real values of $(x, y, z)$ for which

$$
\begin{array}{cccc}
x+y+z & = & 1 \\
x^{2} y+y^{2} z+z^{2} x & = & x y^{2}+y z^{2}+z x^{2} \\
x^{3}+y^{2}+z & = & y^{3}+z^{2}+x
\end{array}
$$

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