A tight fit ....

Geometry Level 5

A circle of (non-zero) radius \(r\) is positioned so that it is tangent to both \(y = \sqrt{x}\) and the positive \(x\)-axis. Another circle, again of the same radius \(r\), is positioned so that it is tangent to both \(y = \sqrt{x}\) and the positive \(y\)-axis.

There is a unique value of \(r\) so that the two circles described above, both of radius \(r\), are also tangent to one another. Find \(\lfloor 10000*r \rfloor\).

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