
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
In the quadrilateral pyramid \( S-ABCD \), the base is a square with a side length of \( 4a \). The lateral edges \( SA = SB = SC = SD = 8a \). \( M \) and \( N \) are the midpoints of edges \( SA \) and \( SC \), respectively. Find the distance \( d \) between the skew lines \( DM \) and \( BN \).

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