Let x, y, z be positive real numbers satisfying 4 x^{2}-2 x y+y^{2}=64, y^{2}-3 y z+3 z^{2}=36, and 4 x^{2}+3 z^{2}=49. If the maximum possible value of 2 x y+y z-4 z x can be expressed as \sqrt{n} for some positive integer n, find n.
Let x, y, z be positive real numbers satisfying 4 x^{2}-2 x y+y^{2}=64, y^{2}-3 y z+3 z^{2}=36, and 4 x^{2}+3 z^{2}=49. If the maximum possible value of 2 x y+y z-4 z x can be expressed as \sqrt{n} for some positive integer n, find n.