
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 real numbers \( x, y, z \) greater than 1 that satisfy the equation

\[ x + y + z + \frac{3}{x - 1} + \frac{3}{y - 1} + \frac{3}{z - 1} = 2(\sqrt{x + 2} + \sqrt{y + 2} + \sqrt{z + 2}). \]

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