
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 sphere is circumscribed around a tetrahedron \(ABCD\). Another sphere with radius 1 is tangent to plane \(ABC\), and both spheres are internally tangent at point \(D\). Given \(AD = 3\), \(\cos \angle BAC = \frac{4}{5}\), and \(\cos \angle BAD = \cos \angle CAD = \frac{1}{\sqrt{2}}\), find the volume of the tetrahedron \(ABCD\).

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