
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 a tetrahedron \(ABCD\), \(AB = m\), \(CD = n\), \(AD = p\), \(BC = q\), \(AC = u\), \(BD = v\). If the angle between \(AB\) and \(CD\) is \(\theta\), then \(\cos \theta = \frac{(p^2 + q^2) - (u^2 + v^2)}{2mn}\).

If the six edge lengths of a tetrahedron are \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\), and the volume is \(V\), then \(a^3 + b^3 + c^3 + d^3 + e^3 + f^3 \geq 36 \sqrt{2}V\).

In a tetrahedron \(ABCD\), if \(AB \perp CD\) and \(AC \perp BD\), then \(AD \perp BC\).

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