
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
Given an equilateral triangle \( \triangle ABC \), let \( D \) be an arbitrary point on side \( BC \). The circumcenter and incenter of \( \triangle ABD \) are \( O_1 \) and \( I_1 \), respectively. The circumcenter and incenter of \( \triangle ADC \) are \( O_2 \) and \( I_2 \), respectively. The lines \( O_1I_1 \) and \( O_2I_2 \) intersect at a point \( P \). Determine the locus of point \( P \) as point \( D \) moves along \( BC \).

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