
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 solutions \( x_{1}, \ldots, x_{5} \) to the system of equations
$$
\left\{\begin{array}{l}
x_{5}+x_{2}=y x_{1}, \\
x_{1}+x_{3}=y x_{2}, \\
x_{2}+x_{4}=y x_{3}, \\
x_{3}+x_{5}=y x_{4}, \\
x_{4}+x_{1}=y x_{3},
\end{array}\right.
$$
where \( y \) is a real parameter.

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