Suppose that x, y, z are nonnegative real numbers satisfying the equation
\sqrt{x y z}-\sqrt{(1-x)(1-y) z}-\sqrt{(1-x) y(1-z)}-\sqrt{x(1-y)(1-z)}=-\frac{1}{2}.
The largest possible value of \sqrt{x y} equals \frac{a+\sqrt{b}}{c}, where a, b, and c are positive integers such that b is not divisible by the square of any prime. Find a^{2}+b^{2}+c^{2}.