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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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