
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 positive integer triples  $(x, y, z) $  that satisfy the equation   $$ x^4+y^4+z^4=2x^2y^2+2y^2z^2+2z^2x^2-63. $$  

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