
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 that skew lines \( a \) and \( b \) are perpendicular to each other, and \( AB \) is the common perpendicular segment with \( A \in a \), \( B \in b \), and \(|AB| = m\). A segment \( PQ \) of fixed length \( n \) has endpoints moving on \( a \) and \( b \), respectively. Let \( N \) and \( M \) be the midpoints of \( PQ \) and \( AB \), respectively. Find the distance \( MN \).

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