
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 nonzero real numbers. Use the listing method to represent the set of all possible values of the algebraic expression \( \frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{xy}{|xy|} + \frac{xyz}{|xyz|} \).

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