Pairwise Cube Sums

Algebra Level 5

$\Large \{6,19,30\}\Longrightarrow6+19=25, 6+30=36, 19+30=49$

Shown above is an example of a set of three numbers such that every pair of numbers in the set sums to the square of a distinct rational number. There are infinitely many of these sets of 3 numbers; unfortunately there are no sets with 4 numbers where this property holds. There are, however, infinitely many sets containing 4 numbers such that each pair of numbers sums to the cube of a distinct rational number. If one of these sets can be written as $$\{w,x,y,z\}$$, where $$w,x,y,z\in\mathbb{Q}$$ and $$w<x<y<z$$, what is the minimum value of $$|w+x+y+z|$$?

×