
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 distinct nonzero real numbers for which the following conditions hold:
\[
x + 2\frac{1}{y} = y + 2\frac{1}{z} = z + 2\frac{1}{x}.
\]
Find the absolute value of the product \(|xyz|\).

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