
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 three non-collinear points $A$, $B$, and $C$ on the same side of plane $E$ such that the plane passing through $A$, $B$, and $C$ is not parallel to plane $E$. Choose arbitrary points $A^{\prime}$, $B^{\prime}$, and $C^{\prime}$ on plane $E$. Let points $L$, $M$, and $N$ be the midpoints of segments $A A^{\prime}$, $B B^{\prime}$, and $C C^{\prime}$ respectively, and assume that $L$, $M$, and $N$ are not collinear. Point $G$ is the centroid of $\triangle L M N$. Determine the locus of point $G$ as $A^{\prime}$, $B^{\prime}$, and $C^{\prime}$ vary arbitrarily on plane $E$.

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