Random reals.

Algebra Level 4

\(a,b,c\) are positive real numbers such that \(\dfrac{1}{b+c} + \dfrac{1}{c+a} + \dfrac{1}{a+b} = 1\). If the minimum value of \(\dfrac{1}{a^5} + \dfrac{1}{b^5} + \dfrac{1}{c^5}\) can be written as \(\dfrac{p}{q}\) where \(p\) and \(q\) are positive integers with \(\text{gcd(p,q)}=1\), find \((p+q)\)

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