
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 rectangular prism has vertices $Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$, and vertices $Q_i$ and $Q_i'$ are opposite each other for $1 \le i \le 4$. The dimensions of the prism are given by lengths 2 along the x-axis, 3 along the y-axis, and 1 along the z-axis. A regular octahedron has one vertex in each of the segments $\overline{Q_1Q_2}$, $\overline{Q_1Q_3}$, $\overline{Q_1Q_4}$, $\overline{Q_1'Q_2'}$, $\overline{Q_1'Q_3'}$, and $\overline{Q_1'Q_4'}$. Find the side length of the octahedron.

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