
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 plane, there are two circles \( C_1 \) and \( C_2 \) with centers \( O_1 \) and \( O_2 \) and radii \( 2R \) and \( R \) respectively \(\left(O_1 O_2 > 3R\right)\). Find the locus of the centroids of triangles, one vertex of which lies on \( C_1 \) and the other two vertices lie on \( C_2 \).

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