
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

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

$$
\begin{aligned}
& 2 y \sqrt{y+1}-z(z+1)=1 \\
& 2 z \sqrt{z+1}-x(x+1)=1
\end{aligned}
$$

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