A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $25$ and $144$ that are externally tangent. For $k\ge 1$, the circles in ${\bigcup }_{j=0}^{k-1}{L}_j$ are ordered according to their points of tangency with the $x$-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 $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S={\bigcup }_{j=0}^5{L}_j$, and for every circle $C$ denote by $r(C)$ its radius. What is ${\sum }_{C\in S}\frac{1}{\sqrt{r(C)}}$?