
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 that $A(x_{A}, y_{A})$ and $B(x_{B}, y_{B})$ are two points symmetric about the origin on the ellipse $C: \frac{x^{2}}{6} + y^{2} = 1$, where $x_{A}x_{B}y_{A}y_{B} \neq 0$, a line $l$ perpendicular to $AB$ is drawn passing through point $A$ and intersecting the ellipse $C$ at points $A$ and $D$. If $k_{AB}$ and $k_{BD}$ represent the slopes of lines $AB$ and $BD$, respectively, then $\frac{k_{AB}}{k_{BD}} = \_\_\_\_\_\_$.

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