
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
Let's consider a cube with vertices at \(Q_1 (0,0,0), Q_2 (0,2,0), Q_3 (0,0,2), Q_4 (0,2,2), Q_1' (2,0,0), Q_2' (2,2,0), Q_3' (2,0,2)\), and \(Q_4' (2,2,2)\). A regular octahedron within the cube 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 \( \overline{Q_1'Q_4'}\) at distances of \(\frac{1}{3}\) of the length of each segment from \(Q_1\) and \(Q_1'\). Find the side length of the octahedron.

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