
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  $O$  be the circumcenter of triangle  $ABC$ . Suppose the perpendicular bisectors of  $\overline{OB}$  and  $\overline{OC}$  intersect lines  $AB$  and  $AC$  at  $D\neq A$  and  $E\neq A$ , respectively.  Determine the maximum possible number of distinct intersection points between line  $BC$  and the circumcircle of  $\triangle ADE$ .

*Andrew Wen*

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