
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 the tetrahedron  $ABCD$  whose edges  $AB$  and  $CD$  have lengths  $a$  and  $b$  respectively. The distance between the skew lines  $AB$  and  $CD$  is  $d$ , and the angle between them is  $\omega$ . Tetrahedron  $ABCD$  is divided into two solids by plane  $\epsilon$ , parallel to lines  $AB$  and  $CD$ . The ratio of the distances of  $\epsilon$  from  $AB$  and  $CD$  is equal to  $k$ . Compute the ratio of the volumes of the two solids obtained.

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