
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
In the upper half-plane of a coordinate plane, there are several circles forming a set. These circles are all tangent to the x-axis. The outermost layer (layer L0) consists of two externally tangent circles with radii 70^2 and 73^2. In ∪_{j=0}^{k-1} L_{j}(k ≥ 1), for every two adjacent circles, a new circle is externally tangent to both. According to this rule, the L_k layer has 2^{k-1} circles. Let S=∪_{j=0}^{6} L_{j}. For each circle C, let its radius be r(C). Find the value of ∑_{C∈S} 1/√(r(C)).

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