
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 \(I\). The circle centered at \(B\) passing through \(I\) intersects side \(AB\) at \(D\), and the circle centered at \(C\) passing through \(I\) intersects side \(AC\) at \(E\). Suppose \(DE\) is the perpendicular bisector of \(AI\). What are all possible measures of angle \(BAC\) in degrees?

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