
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 tetrahedron $ABCD$, $AB=AD=CD=1$, $BD=\sqrt{2}$, and $BD \perp CD$, the plane $ABD \perp$ the plane $BCD$. If the four vertices of the tetrahedron $ABCD$ lie on the same spherical surface, calculate the volume of the sphere.

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