Concentric cubes

Geometry Level 5

There exists a unit cube.

Its midsphere and circumsphere are constructed.

Two new cubes are constructed such that their inspheres are the same as the midsphere and the circumsphere of the first cube.

Then, the midspheres and circumspheres of every existing cube are constructed, and six new cubes are constructed such that their inspheres are the midspheres and circumspheres of the three existing cubes.

After 16 iterations of this process, there will be \(3^{16}\) cubes, including duplicates.

Some of these cubes have no duplicates. What is the sum of all of these unique cubes' side lengths?

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