A collection of circles in the upper half-plane, all tangent to the -axis, is constructed in layers as follows. Layer consists of two circles of radii and that are externally tangent. For , the circles in are ordered according to their points of tangency with the -axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer consists of the circles constructed in this way. Let , and for every circle denote by its radius. What is
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