
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

Consider the system of inequalities $\left\{\begin{array}{l}x + y > 0, \\ x - y < 0\end{array}\right.$ which represents the region $D$ in the plane. A moving point $P$ within region $D$ satisfies the condition that the product of its distances to the lines $x + y = 0$ and $x - y = 0$ is equal to 2. Let the locus of point $P$ be curve $C$. A line $l$ passing through the point $F(2 \sqrt{2}, 0)$ intersects curve $C$ at points $A$ and $B$. If the circle with diameter $\overline{AB}$ is tangent to the $y$-axis, find the slope of line $l$.

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