
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 tetrahedron \(A B C D\) is given. The lengths of the edges \(A B\) and \(C D\) are \(a\) and \(b\), respectively, the distance between the lines \(A B\) and \(C D\) is \(d\), and the angle between them is equal to \(w\). The tetrahedron is divided into two parts by the plane \(\pi\) parallel to the lines \(A B\) and \(C D\). Calculate the ratio of the volumes of the parts if the ratio between the distances of the plane \(\pi\) from \(A B\) and \(C D\) is equal to \(k\).

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