
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 cube \(A B C D A_1 B_1 C_1 D_1\) with edge length \(a\). Let \(M\) be the midpoint of edge \(D_1 C_1\). Find the perimeter of triangle \(A_1 D M\) and the distance from vertex \(D_1\) to the plane passing through the vertices of this triangle.

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