
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 the positive real solution \((x, y, z)\) of the system of equations:
$$
\left\{\begin{array}{l}
2 x^{3}=2 y\left(x^{2}+1\right)-\left(z^{2}+1\right), \\
2 y^{4}=3 z\left(y^{2}+1\right)-2\left(x^{2}+1\right), \\
2 z^{5}=4 x\left(z^{2}+1\right)-3\left(y^{2}+1\right)
\end{array}\right.
$$

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