
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
The numbers \( x \) and \( y \) satisfy the system of equations

$$
\left\{\begin{array}{l}
x + y = a \\
x^{2} + y^{2} = -a^{2} + 2
\end{array}\right.
$$

What is the maximum and minimum value that the product \( x y \) can take?

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