Trigonometry Related Part II

Geometry Level 5

2xy(z+x)(z+y)+2yz(x+y)(x+z)+3zx(y+z)(y+x)\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})}

x,y\color{#3D99F6}{x},\color{#D61F06}{y} and z\color{#20A900}{z} are positive real numbers. If the minimum value of the expression above can be expressed as XY\frac{X}{Y} for positive coprime integers XX and YY, determine X+YX+Y.

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