
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
Find all triples  $(x,y,z)$  of positive integers, with  $z>1$ , satisfying simultaneously that \[x\text{ divides }y+1,\quad y\text{ divides }z-1,\quad z\text{ divides }x^2+1.\]

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