
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
Let the circle \( O: x^{2} + y^{2} = 5 \) intersect the parabola \( C: y^{2} = 2px \) (where \( p > 0 \)) at the point \( A(x_{0}, 2) \). Let \( AB \) be a diameter of the circle \( O \), and a line passing through \( B \) intersects the parabola \( C \) at two distinct points \( D \) and \( E \). Find the product of the slopes of the lines \( AD \) and \( AE \).

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