\[\left ( 1-x \right )^{2}+\left ( x-y \right )^{2}+\left ( y-z \right )^{2}+z^{2}=\frac{1}{4}\] Find all real numbers \( x, y, z\) such that this equation is satisfied. If there exists only one ordered triplet \((x,y,z)\) that satisfies the equation. Then submit your answer as \[\sqrt{\frac{x}{3}}+6y+\sqrt{z}.\]

**Bonus**: Prove that there exists only one ordered triplet of \((x,y,z)\).

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