
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 $P_4'$. Vertices $P_2$, $P_3$, and $P_4$ are adjacent to $P_1$, and vertices $P_i$ and $P_i'$ are opposite each other for $1\le i\le 4$. A regular octahedron has one vertex on 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 $\overline{P_1'P_4'}$. If the vertices on $\overline{P_1P_2}$, $\overline{P_1P_3}$, and $\overline{P_1P_4}$ are $\frac{1}{3}$ of the way from $P_1$ to each respective point, find the side length of the octahedron.

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