
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
Perimeter of triangle  $ABC$  is  $1$ . Circle  $\omega$  touches side  $BC$ , continuation of side  $AB$  at  $P$  and continuation of side  $AC$  in  $Q$ . Line through midpoints  $AB$  and  $AC$  intersects circumcircle of  $APQ$  at  $X$  and  $Y$ .
Find length of  $XY$ . 


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