
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 solutions \( \left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \), where \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) are all positive real numbers, that satisfy:
$$
\begin{array}{l}
\left(x_{1}^{2}-x_{3} x_{3}\right)\left(x_{2}^{2}-x_{3} x_{3}\right) \leqslant 0, \\
\left(x_{3}^{2}-x_{1} x_{1}\right)\left(x_{3}^{2}-x_{1} x_{1}\right) \leqslant 0, \\
\left(x_{3}^{2}-x_{3} x_{2}\right)\left(x_{1}^{2}-x_{3} x_{2}\right) \leqslant 0, \\
\left(x_{1}^{2}-x_{1} x_{3}\right)\left(x_{3}^{2}-x_{1} x_{3}\right) \leqslant 0, \\
\left(x_{3}^{2}-x_{2} x_{1}\right)\left(x_{1}^{2}-x_{2} x_{1}\right) \leqslant 0
\end{array}
$$

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