
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 the systems of values for the unknowns \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) that satisfy the following system of equations, where \( y \) is a parameter:

\[ 
\begin{aligned}
& \text{(1)} \quad x_{5} + x_{2} = y x_{1}, \\
& \text{(2)} \quad x_{1} + x_{3} = y x_{2}, \\
& \text{(3)} \quad x_{2} + x_{4} = y x_{3}, \\
& \text{(4)} \quad x_{3} + x_{5} = y x_{4}, \\
& \text{(5)} \quad x_{4} + x_{1} = y x_{5}.
\end{aligned}
\]

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