
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 of positive real numbers \((x, y, z)\) that satisfy the system

\[ 2 x \sqrt{x+1} - y(y+1) = 1 \]

\[ 2 y \sqrt{y+1} - z(z+1) = 1 \]

\[ 2 z \sqrt{z+1} - x(x+1) = 1 \]

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