
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 incircle of a triangle  $ABC$  touches the side  $AB$  and  $AC$  at respectively at  $X$  and  $Y$ . Let  $K$  be the midpoint of the arc  $\widehat{AB}$  on the circumcircle of  $ABC$ . Assume that  $XY$  bisects the segment  $AK$ . What are the possible measures of angle  $BAC$ ?

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