Largest Inequality Bound

Algebra Level 5

What is the largest integer NN such that (1x2)(1y2)(1z2)x2y2z2>N \frac {(1-x^2)(1-y^2)(1-z^2) } {x^2 y^2 z^2} > N , as x,yx, y and zz range over all positive reals subject to x+y+z<1x + y + z < 1 ?

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