
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
Three mutually perpendicular rays  $O_x,O_y,O_z$  and three points  $A,B,C$  on  $O_x,O_y,O_z$ , respectively. A variable sphere є through  $A, B,C$  meets  $O_x,O_y,O_z$  again at  $A', B',C'$ , respectively. Let  $M$  and  $M'$  be the centroids of triangles  $ABC$  and  $A'B'C'$ . Find the locus of the midpoint of  $MM'$ .

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