
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  $ABC$  be a triangle with incenter  $I$  and incircle  $\omega$ .  It is given that there exist points  $X$  and  $Y$  on the circumference of  $\omega$  such that  $\angle BXC=\angle BYC=90^\circ$ .  Suppose further that  $X$ ,  $I$ , and  $Y$  are collinear.  If  $AB=80$  and  $AC=97$ , compute the length of  $BC$ .

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