
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 set of circles in a plane, all tangent to the $x$-axis, is arranged in layers. The first layer, $L_0$, contains two circles with radii $50^2$ and $54^2$ that are externally tangent. For $k \geq 1$, the circles in $\bigcup_{j=0}^{k-1} L_j$ are ordered by their tangency points on the $x$-axis. For every pair of consecutive circles, a new circle is constructed that is externally tangent to each circle in the pair as well as tangent to the $x$-axis. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this manner. Define $S = \bigcup_{j=0}^5 L_j$. For each circle $C$ in $S$, let $r(C)$ be its radius. Calculate the sum:
\[
\sum_{C \in S} \frac{1}{\sqrt{r(C)}}
\]
A) $\frac{243}{54}$
B) $\frac{250}{54}$
C) $\frac{251}{54}$
D) $\frac{252}{54}$
E) $\frac{253}{54}$

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