
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 regular tetrahedron \(ABCD\) with edge length \(a\), point \(M\) is the midpoint of \(AB\). Find the angle and distance between the lines \(AD\) and \(CM\). In what ratio does the common perpendicular of these lines divide the segments \(CM\) and \(AD\)?

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