
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
Solve the system of equations:
\[
\begin{cases}
x^{2} - y^{2} + z = \frac{27}{xy} \\
y^{2} - z^{2} + x = \frac{27}{yz} \\
z^{2} - x^{2} + y = \frac{27}{xz}
\end{cases}
\]

### Answer
\((3, 3, 3), (-3, -3, 3), (-3, 3, -3), (3, -3, -3)\)

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