
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
A quadrangular prism \(A B C D A_{1} B_{1} C_{1} D_{1}\) has a rhombus \(A B C D\) at its base, where \(A C = 4\) and \(\angle D B C = 30^{\circ}\). A sphere passes through the vertices \(D, A, B, B_{1}, C_{1}, D_{1}\).

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

b) Find the angle \(A_{1} C D\).

c) It is additionally given that the radius of the sphere is 5. Find the volume of the prism.

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