
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 triples \((x, y, z)\) of positive integers such that

\[ 
x \mid (y+1), \quad y \mid (z+1), \quad \text{and} \quad z \mid (x+1) 
\]

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