
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
Given a rectangular parallelepiped with edges \(a < b < c\), consider cubes whose centers coincide with the center of symmetry of the parallelepiped and whose faces are parallel to the faces of the parallelepiped. Find the edge length of the cube that minimizes the difference between the volumes of the union and the intersection with this parallelepiped.

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