
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 the quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is a rhombus $A B C D$, where $B D=3$ and $\angle A D C=60^{\circ}$. A sphere passes through the vertices $D, C, B, B_{1}, A_{1}, D_{1}$.

a) Find the area of the circle obtained in the cross-section of the sphere by the plane passing through the points $A_{1}, C_{1}$, and $D_{1}$.

b) Find the angle $B_{1} C_{1} A$.

c) It is additionally known that the radius of the sphere is 2. Find the volume of the prism.

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