\[\Large \frac{2\color{blue}{x}\color{red}{y}}{(\color{green}{z}+\color{blue}{x})(\color{green}{z}+\color{red}{y})}+\frac{2\color{red}{y}\color{green}{z}}{(\color{blue}{x}+\color{red}{y})(\color{blue}{x}+\color{green}{z})}+\frac{3\color{green}{z}\color{blue}{x}}{(\color{red}{y}+\color{green}{z})(\color{red}{y}+\color{blue}{x})}\]

\(\color{blue}{x},\color{red}{y}\) and \(\color{green}{z}\) are positive real numbers. If the minimum value of the expression above can be expressed as \(\frac{X}{Y}\) for positive coprime integers \(X\) and \(Y\), determine \(X+Y\).

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