
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 trapezoid \( KLMN \), diagonal \( KM \) is equal to 1 and is also its height. From points \( K \) and \( M \), perpendiculars \( KP \) and \( MQ \) are drawn to sides \( MN \) and \( KL \), respectively. Find \( LM \) if \( KN = MQ \) and \( LM = MP \).

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