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Algebra Level 4

Let \(x,y,z\) be real numbers satisfying \(x+y+z = 0\) and \(x^2 + y^2 + z^2 = 8\). If the minimum value of \(|x| + |y| + |z|\) is \(S\), find \(\left\lfloor 10000S \right\rfloor\).

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