
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 rectangular prism \( ABCD-A_1B_1C_1D_1 \) with \( AB = 4 \), \( AD = 3 \), and \( AA_1 = 2 \), let \( M \) and \( N \) be the midpoints of \( DC \) and \( BB_1 \), respectively. Find the distance between the skew lines \( MN \) and \( A_1B \).

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