
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
Consider a unit cube with vertices $V_1, V_2, V_3, V_4, V_1', V_2', V_3', V_4'$ such that vertices $V_2, V_3, V_4$ are adjacent to $V_1$, and for $1 \leq i \leq 4,$ vertices $V_i$ and $V_i'$ are opposite to each other. A regular octahedron is positioned such that it has one vertex at $(x, 0, 0)$ on the edge $\overline{V_1 V_2}$, one vertex at $(0, y, 0)$ on the edge $\overline{V_1 V_3}$, one vertex at $(0, 0, z)$ on the edge $\overline{V_1 V_4}$, and corresponding opposite vertices along $\overline{V_1' V_2'}, \overline{V_1' V_3'}, \overline{V_1' V_4'}$ are at $(1, 1-y, 1)$, $(1-x, 1, 1)$, and $(1, 1, 1-z)$. If the octahedron vertices divide each edge in the ratio 3:1, identify the side length of the octahedron.

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