
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
Let  $O$  and  $I$  be the circumcenter and incenter of triangle  $ABC$ . The perpendicular from  $I$  to  $OI$  meets  $AB$  and the external bisector of angle  $C$  at points  $X$  and  $Y$  respectively. In what ratio does  $I$  divide the segment  $XY$ ?

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