There was this dog named Rita

Consider the equation

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

where a,b,c,da, b, c, d are distinct positive integers, at least 33 of which are prime.

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

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