
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 a unit cube \( ABCD-A_1B_1C_1D_1 \), \( M \) and \( N \) are the midpoints of \( BB_1 \) and \( B_1C_1 \) respectively, and \( P \) is the midpoint of line segment \( MN \). Find the distance between \( DP \) and \( AC_1 \).

(1982 Shanghai High School Competition Problem)

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