Geometry, Algebra, Combinatorics, MIXED?

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

1. $C_{0}$ is the circle $x^{2} + y^{2} = 1$
2. 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.