Trigonometry Related Part II

Geometry Level 5

$\Large \frac{2\color{#3D99F6}{x}\color{#D61F06}{y}}{(\color{#20A900}{z}+\color{#3D99F6}{x})(\color{#20A900}{z}+\color{#D61F06}{y})}+\frac{2\color{#D61F06}{y}\color{#20A900}{z}}{(\color{#3D99F6}{x}+\color{#D61F06}{y})(\color{#3D99F6}{x}+\color{#20A900}{z})}+\frac{3\color{#20A900}{z}\color{#3D99F6}{x}}{(\color{#D61F06}{y}+\color{#20A900}{z})(\color{#D61F06}{y}+\color{#3D99F6}{x})}$

$\color{#3D99F6}{x},\color{#D61F06}{y}$ and $\color{#20A900}{z}$ 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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