
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 $A_1, A_2, A_3, A_4, A_1', A_2', A_3'$, and $A_4'$. Vertices $A_2$, $A_3$, and $A_4$ are adjacent to $A_1$, and for $1\le i\le 4,$ vertices $A_i$ and $A_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $\overline{A_1A_2}$, $\overline{A_1A_3}$, $\overline{A_1A_4}$, $\overline{A_1'A_2'}$, $\overline{A_1'A_3'}$, and $\overline{A_1'A_4'}$. Find the side length of the octahedron if the vertices of the octahedron on $\overline{A_1A_2}$ and $\overline{A_1'A_2'}$ are $\frac{1}{3}$rd and $\frac{2}{3}$rd along the segments respectively.

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