
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
Triangle  $ABC$  has an incenter  $I$ . Let points  $X$ ,  $Y$  be located on the line segments  $AB$ ,  $AC$  respectively, so that  $BX \cdot AB = IB^2$  and  $CY \cdot AC = IC^2$ . Given that the points  $X, I, Y$  lie on a straight line, find the possible values of the measure of angle  $A$ .

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