
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
Two non-rolling circles  $C_1$  and  $C_2$  with centers  $O_1$  and  $O_2$  and radii  $2R$  and  $R$ , respectively, are given on the plane. Find the locus of the centers of gravity of triangles in which one vertex lies on  $C_1$  and the other two lie on  $C_2$ .

 (B. Frenkin)

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