
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
Let \( O \) be a point in three-dimensional space, and let \( l_1, l_2, l_3 \) be mutually perpendicular straight lines passing through \( O \). Let \( S \) denote the sphere with center \( O \) and radius \( R \). For every point \( M \) on \( S \), let \( S_M \) denote the sphere with center \( M \) and radius \( R \). We denote by \( P_1, P_2, P_3 \) the intersection of \( S_M \) with the straight lines \( l_1, l_2, l_3 \), respectively. Define \( P_1 \neq O \) if \( l_1 \) meets \( S_M \) at two distinct points and \( P_1 = O \) otherwise (similarly for \( P_2 \) and \( P_3 \)). What is the set of centers of gravity of the (possibly degenerate) triangles \( P_1 P_2 P_3 \) as \( M \) runs through the points of \( S \)?

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