
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
Solve the following system of equations.

$$
\begin{gathered}
\sqrt{x}+\sqrt{y}=z \\
2 x+2 y+p=0 \\
z^{4}+p z^{2}+q=0
\end{gathered}
$$

Under what conditions do the real values of \( x \), \( y \), and \( z \) satisfy the system of equations?

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