
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 the rectangular parallelepiped \( A B C D A_1 B_1 C_1 D_1 \), it is known that \( A B = 3 \), \( B C = 2 \), and \( C C_1 = 4 \). On the edge \( A B \), there is a point \( M \) such that \( A M : M B = 1 : 2 \); \( K \) is the intersection point of the diagonals of the face \( C C_1 D_1 D \). Find the angle and the distance between the lines \( D_1 M \) and \( B_1 K \).

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