
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 pair of circles in the upper half-plane, both tangent to the x-axis, consists of one circle with radius $49$ and another circle with radius $64$. A new circle is constructed externally tangent to both of these initial circles. Determine the value of:
\[
\frac{1}{\sqrt{r_1}} + \frac{1}{\sqrt{r_2}} + \frac{1}{\sqrt{r_3}}
\]
where \( r_1 = 49 \), \( r_2 = 64 \), and \( r_3 \) is the radius of the newly constructed circle.
A) \(0.573\)
B) \(0.535867218\)
C) \(0.425\)
D) \(0.789\)

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