
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 unit cube has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3',$ and $P_4'$, with vertices $P_1$, $P_3$, and $P_4$ adjacent to $P_2$, and for $1 \le i \le 4$, vertices $P_i$ and $P_i'$ are opposite each other. A regular octahedron has its vertices located such that each vertex lies on one of the segments $\overline{P_2P_1}$, $\overline{P_2P_3}$, $\overline{P_2P_4}$, $\overline{P_2'P_1'}$, $\overline{P_2'P_3'}$, and $\overline{P_2'P_4'}$. Determine the side length of this octahedron.

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