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 \(|w+x+y+z|\)?

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