
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
In the regular quadrilateral pyramid \( S \) \( A \) \( B \) \( C \) \( D \), the edge \( A \) \( B \) is twice the height of the pyramid. On one side of the plane of the base \( A \) \( B \) \( C \) \( D \), there is a cylinder with its base circle passing through the center of the base \( A \) \( B \) \( C \) \( D \). The orthogonal projections of the cylinder on the planes \( S \) \( C \) \( D \) and \( S \) \( B \) \( C \) are rectangles with a common vertex at point \( C \). Find the ratio of the volumes of the cylinder and the pyramid.

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