
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 a point on the edge $A_1 D_1$ such that $A_1 M : M D_1 = 1 : 2$. Find the perimeter of the triangle $A B_1 M$, as well as the distance from the 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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