
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 \( ABCDA_1B_1C_1D_1 \) with edge length \( a \). Let \( M \) be a point on the edge \( A_1D_1 \) such that \( A_1M : MD_1 = 1 : 2 \). Find the perimeter of triangle \( AB_1M \), as well as the distance from vertex \( A_1 \) to the plane passing through the vertices of this triangle.

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