
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
Two circles in $L_0$, tangent to the $x$-axis, have radii 50^2 and 53^2. For any set of two adjacent circles in a layer, a new circle in the subsequent layer is constructed externally tangent to both. Starting from two externally tangent circles, subsequent layers continue to fill in new circles between existing pairs. Compute the sum of the reciprocals of the square roots of the radii of all circles up to and including layer $L_6$.
**A)** $\frac{13021}{2650}$ 
**B)** $\frac{2650}{13021}$ 
**C)** $\frac{100}{103}$ 
**D)** $\frac{384}{79}$

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