
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 solutions for the system of equations:
\[
\begin{cases}
\left(x_3 + x_4 + x_5\right)^5 = 3x_1, \\
\left(x_4 + x_5 + x_1\right)^5 = 3x_2, \\
\left(x_5 + x_1 + x_2\right)^5 = 3x_3, \\
\left(x_1 + x_2 + x_3\right)^5 = 3x_4, \\
\left(x_2 + x_3 + x_4\right)^5 = 3x_5.
\end{cases}
\]

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