
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:
$$
\begin{gathered}
x+y=a(\sqrt{x}-\sqrt{y}) \\
x^{2}+y^{2}=a(\sqrt{x}-\sqrt{y})^{2}
\end{gathered}
$$
where \( a \) is a real number. For which values of \( a \) will the roots be real?

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