
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A configuration of circles in the upper half-plane, all tangent to the $x$-axis, is arranged in layers as follows. Layer \(L_0\) consists of two circles with radii \(60^2\) and \(65^2\), positioned such that they are externally tangent. For any \(k \ge 1\), the circles in \(\bigcup_{j=0}^{k-1}L_j\) are sorted by their tangency points on the $x$-axis. For each adjacent circle pair, a new circle is constructed externally tangent to both circles in the pair. Layer \(L_k\) includes the \(2^{k-1}\) newly constructed circles. Define \(S=\bigcup_{j=0}^{5}L_j\). For every circle \(C\) in \(S\), denote by \(r(C)\) its radius. Calculate:
\[
\sum_{C\in S} \frac{1}{\sqrt{r(C)}}
\]
A) $\frac{105}{52}$  
B) $\frac{210}{103}$  
C) $\frac{320}{159}$  
D) $\frac{420}{203}$  
E) $\frac{625}{301}$

Present the answer in LaTex format: \boxed{Your answer}
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