
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  $(x, y,z)$  consisting of three distinct real numbers, that satisfy the following system of equations:
  $\begin {cases}x^2 + y^2 = -x + 3y + z 
y^2 + z^2 = x + 3y - z 
x^2 + z^2 = 2x + 2y - z \end {cases}$ 

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