Find the smallest positive integer G such that there exist distinct positive integers a, b, c with the following properties:
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\operatorname{gcd}(a, b, c)=G.
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\operatorname{lcm}(a, b)=\operatorname{lcm}(a, c)=\operatorname{lcm}(b, c).
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\frac{1}{a}+\frac{1}{b}, \frac{1}{a}+\frac{1}{c}, and \frac{1}{b}+\frac{1}{c} \text{ are reciprocals of integers.}
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\operatorname{gcd}(a, b)+\operatorname{gcd}(a, c)+\operatorname{gcd}(b, c)=16 G.