
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 $A, B, C, D, A', B', C',$ and $D'$, where $B$, $C$, and $D$ are adjacent to $A$, and for $1 \le i \le 4,$ vertices $A_i$ and $A_i'$ are opposite each other. A regular octahedron has one vertex in each of the segments $\overline{A B}$, $\overline{A C}$, $\overline{A D}$, $\overline{A' B'}$, $\overline{A' C'}$, and $\overline{A' D'}$ with a distance of $1/3$ from $A$ or $A'$ to the vertex on each segment.  Find the side length of the octahedron.

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