Cyclically 130

Algebra Level 5

Consider all quadruples of real numbers \((x_1,x_2,x_3,x_4),\) satisfying \(x_1\leq x_2 \leq x_3 \leq x_4\), such that

\[ \begin{cases} x_1 + x_2x_3x_4 & = 130\\ x_2 + x_3x_4x_1 & = 130\\ x_3 + x_4x_1x_2 & = 130\\ x_4 + x_1x_2x_3 & = 130\\ \end{cases} \]

If \(S\) is the sum of all possible values of \(x_4,\) what is \(S,\) rounded to the nearest integer?

Details and assumptions

Each possible distinct value of \(x_4\) should be counted only once, even if it appears in several different quadruples.

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