
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 the equilateral  $\bigtriangleup ABC$ ,  $D$  is a point on side  $BC$ .   $O_1$  and  $I_1$  are the circumcenter and incenter of  $\bigtriangleup ABD$  respectively, and  $O_2$  and  $I_2$  are the circumcenter and incenter of  $\bigtriangleup ADC$  respectively.   $O_1I_1$  intersects  $O_2I_2$  at  $P$ .  Find the locus of point  $P$  as  $D$  moves along  $BC$ .

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