
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
Given plane $\varepsilon$ and on one side of the plane points $A, B, C$ which are not collinear and the plane determined by them is not parallel to $\varepsilon$. Let $A^{\prime}, B^{\prime}, C^{\prime}$ be any three points on the plane $\varepsilon$. The midpoints of the segments $A A^{\prime}, B B^{\prime}, C C^{\prime}$ are $L, M$, and $N$ respectively, and the centroid of triangle $L M N$ is $G$. (We ignore such triples of points $A^{\prime}, B^{\prime}, C^{\prime}$ for which $L, M, N$ do not form a triangle.) What is the locus of points $G$ as $A^{\prime}, B^{\prime}, C^{\prime}$ independently run through the plane $\varepsilon$?

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