
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 the triangular prism \(A-BCD\), the side edges \(AB, AC, AD\) are mutually perpendicular. The areas of triangles \(\triangle ABC\), \(\triangle ACD\), and \(\triangle ADB\) are \(\frac{\sqrt{2}}{2}\), \(\frac{\sqrt{3}}{2}\), and \(\frac{\sqrt{6}}{2}\) respectively. Find the volume of the circumscribed sphere of the triangular prism \(A-BCD\).

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