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Let \(a > b > c \; \) be the roots of the equation \(\begin{align} (x-1)^3 + (x+3)^3 = 42(x+1) \end{align}\)

The value of

\[\begin{align} \dfrac{\left (\dfrac{a+b}{a^a+b^b} \right )^{(a+\sqrt{bc}+c)}}{\left (\dfrac{1}{a+b}+\dfrac{1}{a+c}+\dfrac{1}{b+c} \right )^{(a-b+c)}} \end{align}\]

can be written as \(\dfrac{m}{n}\), where \(m\) and \(n\) are positive coprime numbers. Evaluate \(\begin{align} \dfrac{\sqrt{m^m+n^m-m}}{3m^m-n^2+20}.\end{align}\)

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