
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 triangle  $ABC$  with  $AB=8$  and  $AC=10$ , the incenter  $I$  is reflected across side  $AB$  to point  $X$  and across side  $AC$  to point  $Y$ . Given that segment  $XY$  bisects  $AI$ , compute  $BC^2$ . (The incenter  is the center of the inscribed circle of triangle .)

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