
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
Let \( x, y, z \) be real numbers not equal to zero such that

\[ \frac{x+y}{z} = \frac{y+z}{x} = \frac{z+x}{y} .\]

Determine all possible values of

\[ \frac{(x+y)(y+z)(z+x)}{xyz}. \]

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