
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
The circles with centers \( O_{1} \) and \( O_{2} \) touch each other externally at point \( E \), and \( E O_{1} = 2 \cdot E O_{2} \). A line rotating around \( E \) intersects the circles again at points \( M_{1} \) and \( M_{2} \), respectively. What is the locus of the intersection point of the lines \( M_{1} O_{2} \) and \( M_{2} O_{1} \)?

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