
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
From one vertex of a parallelepiped - let this be $O$ - consider the 3 edge vectors denoted by $a$, $b$, and $c$. Find the midpoints of the three faces passing through the opposite vertex, and then find the centroid of the triangle formed by these midpoints. Express the vector from $O$ to the centroid in terms of $a$, $b$, and $c$.

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