Let \(C_{0}, C_{1}, C_{2},...\) be a sequence of circles in the Cartesian plane defined as:

- \(C_{0}\)is the circle \(x^{2} + y^{2} = 1\)
- For \(n = 0, 1, 2, 3, ...\), the circle \(C_{n+1}\) lies in the upper half plane and is tangent to \(C_{n}\) as well as both branches of the hyperbola \(x^{2} - y^{2} = 1\)

Let \(r_{n}\) be the radius of \(C_{n}\). Find the length of \(r_{10}\)

Please use Wolfram Alpha only in the last step. This is a modified APMO problem.

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