
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 perimeter of triangle \( \mathrm{ABC} \) is 1. A circle touches side \( \mathrm{AB} \) at point \( P \) and the extension of side \( \mathrm{AC} \) at point \( Q \). A line passing through the midpoints of sides \( \mathrm{AB} \) and \( \mathrm{AC} \) intersects the circumcircle of triangle \( \mathrm{APQ} \) at points \( X \) and \( Y \). Find the length of segment \( X Y \).

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