
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
ABCD and A1B1C1D1 form a cube. \(AB = 1\); \( \overline{DM} = \frac{8}{15} \overline{DB1}\). The plane \(\alpha\) passes through point \(M\) and is perpendicular to line \(B1D\). Find how the plane \(\alpha\) divides edge \(A1D1\).

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