
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
Given a tetrahedron \(ABCD\) with edge \(AB\) of length \(a\) and edge \(CD\) of length \(b\). The distance between the skew lines of \(AB\) and \(CD\) is \(d\), and the angle between them is \(\omega\). A plane \(\varepsilon\) parallel to edges \(AB\) and \(CD\) separates the tetrahedron into two parts. What is the ratio of the volumes of these two parts if the distance from the line \(AB\) to the plane \(\varepsilon\) is \(k\) times the distance from the line \(CD\) to the plane \(\varepsilon\)?

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