
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, \(AC_1\) is a diagonal. \(M\) and \(N\) are the midpoints of \(BB_1\) and \(B_1C_1\) respectively. \(P\) is the midpoint of line segment \(MN\). Find the distance between the skew lines \(DP\) and \(AC_1\).

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