
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 a real number $p>0$, the line $3x-4y+2p=0$ intersects the parabola $x^{2}=2py$ and the circle $x^{2}+\left(y-\frac{p}{2}\right)^{2}= \frac{p^{2}}{4}$ at four points from left to right: A, B, C, D. Calculate the ratio $\frac{AB}{CD}$.

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