
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 cuboid has dimensions of 2 units by 2 units by 2 units. It has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3', P_4'.$ Vertices $P_2, P_3,$ and $P_4$ are adjacent to $P_1$, and vertices $P_i' (i = 1,2,3,4)$ are opposite to $P_i$. A regular octahedron has one vertex in each of the segments $\overline{P_1P_2}, \overline{P_1P_3}, \overline{P_1P_4}, \overline{P_1'P_2'}, \overline{P_1'P_3'},$ and $\overline{P_1'P_4'}$ with each vertex positioned $\frac{2}{3}$ of the distance from $P_1$ or $P_1'$.

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