
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
Given an ellipse $C$: $\frac{x^2}{8}+\frac{y^2}{4}=1$ with the left and right foci $F_{1}$ and $F_{2}$ respectively. A line $l$ passing through the right focus $F_{2}$ intersects $C$ at points $A$ and $B$. If the center of the circle passing through points $A$, $B$, and $(-\sqrt{7},0)$ lies on the $y$-axis, then the slope of line $l$ is ______.

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