A geometric inequality?

Algebra Level 4

\[{\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+x)(y+z)}+\frac{z^2}{(z+x)(z+y)}<A}\]

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

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