
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 a regular quadrangular pyramid \( S A B C D \), the height is equal to the diagonal of the base \( A B C D \). Through vertex \( A \), a plane parallel to line \( B D \) is drawn, tangent to the sphere inscribed in the pyramid. Find the ratio of the cross-sectional area to the base area of the pyramid.

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