
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 all positive real solutions to the following system of inequalities:

\[
(x_1^2 - x_3 x_5)(x_2^2 - x_3 x_5) \leq 0
\]
\[
(x_2^2 - x_4 x_1)(x_3^2 - x_4 x_1) \leq 0
\]
\[
(x_3^2 - x_5 x_2)(x_4^2 - x_5 x_2) \leq 0
\]
\[
(x_4^2 - x_1 x_3)(x_5^2 - x_1 x_3) \leq 0
\]
\[
(x_5^2 - x_2 x_4)(x_1^2 - x_2 x_4) \leq 0
\]

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