
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 of the product \( xy \)?

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