
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
Consider a set of circles in the upper half-plane, all tangent to the $x$-axis. Begin with two circles of radii $50^2$ and $53^2$, externally tangent to each other, defined as Layer $L_0$. For each pair of consecutive circles in $\bigcup_{j=0}^{k-1}L_j$, a new circle in Layer $L_k$ is constructed externally tangent to each circle in the pair. Construct this way up to Layer $L_3$. Let $S=\bigcup_{j=0}^{3}L_j$. For each circle $C$ denote by $r(C)$ its radius. Find
\[
\sum_{C \in S} \frac{1}{{r(C)}}.
\]
A) $\frac{1}{1450}$
B) $\frac{1}{1517}$
C) $\frac{1}{1600}$
D) $\frac{1}{1700}$

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