Cyclically 130

Algebra Level 5

Consider all quadruples of real numbers (x1,x2,x3,x4),(x_1,x_2,x_3,x_4), satisfying x1x2x3x4x_1\leq x_2 \leq x_3 \leq x_4, such that

{x1+x2x3x4=130x2+x3x4x1=130x3+x4x1x2=130x4+x1x2x3=130 \begin{cases} x_1 + x_2x_3x_4 & = 130\\ x_2 + x_3x_4x_1 & = 130\\ x_3 + x_4x_1x_2 & = 130\\ x_4 + x_1x_2x_3 & = 130\\ \end{cases}

If SS is the sum of all possible values of x4,x_4, what is S,S, rounded to the nearest integer?

Details and assumptions

Each possible distinct value of x4x_4 should be counted only once, even if it appears in several different quadruples.

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