
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 \(ABCD-A_1B_1C_1D_1\) with side length 1, and \(E\) as the midpoint of \(D_1C_1\), find the following:
1. The distance between skew lines \(D_1B\) and \(A_1E\).
2. The distance from \(B_1\) to plane \(A_1BE\).
3. The distance from \(D_1C\) to plane \(A_1BE\).
4. The distance between plane \(A_1DB\) and plane \(D_1CB_1\).

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