
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
A plane \(\epsilon\) is given, and on one side of the plane are three non-collinear points \(A\), \(B\), and \(C\) such that the plane determined by them is not parallel to \(\epsilon\). Three arbitrary points \(A', B'\), and \(C'\) in \(\epsilon\) are selected. Let \(L, M\), and \(N\) be the midpoints of \(AA', BB'\), and \(CC'\), respectively, and let \(G\) be the centroid of \(\triangle LMN\). Find the locus of all points obtained for \(G\) as \(A', B'\), and \(C'\) are varied independently of each other across \(\epsilon\).

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