\[(x+y+z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \ge s \left( \frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} \right)\]

\(x,y,z\) are real numbers in the interval \([1, 2]\). Let \(S\) be the maximum value of \(s\) such that the inequality holds for all such \(x,y,z\), and suppose that in this case, equality is achieved when \(Hx = Ky = Az\). Compute \(S+H+K+A\).

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