\[\frac{4(a+c)}{a^2+3c^2+28}+\frac{4a}{a^2+bc+7}-\frac{5}{(a+b)^2}-\frac{3}{a(b+c)}\]

Given that \(a,b,c\) are positive reals and \(a^2+b^2+c^2=14\).If the maximum value of the expression above can be expressed as \(\large \frac{H}{K}\) and the sum of \(a,b,c\) when the equality holds is \(\large A\).

Compute \(\large H+K+A\).

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