
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 a radius of 1 is tangent to the plane \(ABC\), and both spheres are tangent to point \(D\). Given the conditions 
\[
\begin{array}{l}
A D=3, \cos \angle BAC=\frac{4}{5}, \\
\cos \angle BAD=\cos \angle CAD=\frac{1}{\sqrt{2}},
\end{array}
\]
what is the volume of tetrahedron \(ABCD\)?

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