Minimal Value

Algebra Level 3

Given positive reals \(a,b,c\) and \(ab+bc+ca\ge 3\). The minimum value of \(\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b}}\) is \(\frac{m}{\sqrt{n}}\), where \(m,n\) are square-free integers. Find the value of \(mn\).

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