
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
The base of a quadrilateral prism \( A B C D A_1 B_1 C_1 D_1 \) is a rhombus \( A B C D \) with \( CD = 3 \) and \( \angle A B D = 30^\circ \). A sphere passes through the vertices \( D, C, B, B_1, A_1, D_1 \).

a) Find the area of the circle obtained by the intersection of the sphere with the plane passing through points \( A, C \), and \( D \).

b) Find the angle \( C_1 A B \).

c) Given that the radius of the sphere is 6, find the volume of the prism.

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