
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 cube \( ABCD-A_1 B_1 C_1 D_1 \) with edge length 1, points \( X \) and \( Y \) are the centers of the squares \( AA_1 B_1 B \) and \( BB_1 C_1 C \) respectively. Point \( Z \) lies on the diagonal \( BD \) such that \( DZ = 3 ZB \). Determine the area of the cross-section cut by the plane \( XYZ \) on the circumscribed sphere of the cube.

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