\[\large x^5+y^5+z^5\]

Given that \(x,y\) and \(z\) are real numbers satisfying \(x+y+z=0 \) and \(x^2+y^2+z^2 = 1\). And the maximum value of of the expression above can be expressed as \( \dfrac{ H\sqrt K}A \), where \(A,H\) and \(K \) are positive integers with \(K\) square-free and \(A,H\) copime. Find \(H+K+A\).

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