Consider the equation

\[\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = \dfrac{1}{d} \]

where \(a, b, c, d\) are distinct positive integers, at least \(3\) of which are prime.

Up to a permutation of the variables, there is a unique solution to this problem. Find \(a + b + c + d\).

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