
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 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 edge \(A B\), point \(M\) is taken such that \(A M : M B = 1 : 2\). Let \(K\) be the point of intersection of the diagonals of face \(C C_1 D_1 D\). Find the angle and the distance between lines \(D_1 M\) and \(B_1 K\).

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