
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 $e_{1}, e_{2}$ and $e_{3}$ be three skew edges of a cube. Determine the set of centroids of the triangles $E_{1} E_{2} E_{3}$, where $E_{i}$ is a point on the edge $e_{i}$ $(i=1,2,3)$.

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