
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
The length of the edge of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is $a$. Points $P, K, L$ are the midpoints of edges $A A_{1}, A_{1} D_{1}, B_{1} C_{1}$ respectively, and point $Q$ is the center of face $C C_{1} D_{1} D$. The segment $M N$ with endpoints on lines $A D$ and $K L$ intersects the line $P Q$ and is perpendicular to it. Find the length of this segment.

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