Almost A Sphere, But Not Quite

Algebra Level 5

Let x,y,zx,y,z be real numbers such that x2+y2+z2+(x+y+z)2=9x^2+y^2+z^2+(x+y+z)^2=9 and xyz1532xyz\leq \frac{15}{32}. To 2 decimal places, what is the greatest possible value of xx?

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