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Algebra Level 5

\[\dfrac{1}{a^2+b^2}+\dfrac{1}{b^2+c^2}+\dfrac{1}{c^2+a^2}+10\sqrt{(a+1)(b+1)(c+1)}\]

Let \(a,b\) and \(c\) be positive reals satisfying \(ab+bc+ca=1\). If the minimum value of the expression above be denoted as \(S\), compute \( \dfrac{896 S}{10} \).

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