
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
Let $ABCD$ be a tetrahedron where edges $AB$, $AC$, and $AD$ are mutually perpendicular. Assume triangles $ABC$, $ACD$, and $ADB$ have areas denoted by $w$, $v$, and $u$, respectively. Let $BC = 2w$, $CD = 2v$, and $BD = 2u$. Find the volume of the tetrahedron $ABCD$.

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