A geometric inequality?

Algebra Level 4

x2(x+y)(x+z)+y2(y+x)(y+z)+z2(z+x)(z+y)<A{\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+x)(y+z)}+\frac{z^2}{(z+x)(z+y)}<A}

The above inequality is true for any positive numbers x,y,x, y, and zz. Find the smallest number A,A, and prove that your answer is right.

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