
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  $(x,y,z)$  which satisfy the equations: $1)$   $x(y^2+z)=z(z+xy)$  $2)$   $y(z^2+x)=x(x+yz)$  $3)$   $z(x^2+y)=y(y+xz)$ 

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