A unit disk (radius = 1) is partitioned into an inner circular region of radius \((r < 1)\) and an outer region consisting of the area between concentric circles of radii \(r\) and 1 respectively.

The inner region is preserved, and the outer region is reformed into a circular disk with equivalent area. Thus, the combined area of the two resulting disks is equal to the area of the original unit disk.

Which value of \(r\) (to 3 decimal places) maximizes the sum of the circumferences of the two final disks?

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