
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 cube with side length 2 has vertices \( Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3', Q_4' \). Vertices \( Q_2, Q_3, \) and \( Q_4 \) are adjacent to \( Q_1 \), and for \( 1 \leq i \leq 4 \), vertices \( Q_i \) and \( Q_i' \) are opposite to each other. 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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