
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 tetrahedron \(ABCD\) has the following properties: \(AB = 7\), \(DC = 8\), \(\angle BCD = \angle ADC = 75^\circ\), and \(\angle BDC = \angle ACD = 15^\circ\). A sphere is circumscribed around the tetrahedron. Consider the set of all points on this sphere such that the sum of the spherical distances from these points to the vertices \(A, B, C, D\) is no more than \(8\pi\). What is the area of this set? The spherical distance between two points on a sphere is the length of the shortest arc of a great circle connecting them.

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