There was this dog named Rita

Number Theory Level 5

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 $a + b + c + d$ .