Suppose that a $3-$way switch is designed such that

If all three UP or all three DOWN, then light is ON

If exactly one UP and other two DOWN, then light is ON

All other combinations keep the lights OFF

Which of the following logic circuits best describes the functionality of a $3-$way switch with two switches $x,y$ and $z?$ We denote the negation of $x$ by $x^{\text{'}}$

$xyz+x^{\text{'}}{y}^{\text{'}}{z}^{\text{'}}+x^{\text{'}}y{z}^{\text{'}}+x^{\text{'}}{y}^{\text{'}}z$

$xyz+x^{\text{'}}{y}^{\text{'}}{z}^{\text{'}}+x^{\text{'}}{y}^{\text{'}}z$

$xyz+x{y}^{\text{'}}{z}^{\text{'}}+x^{\text{'}}y{z}^{\text{'}}+x^{\text{'}}{y}^{\text{'}}$

$xyz+x^{\text{'}}{y}^{\text{'}}{z}^{\text{'}}+x{y}^{\text{'}}{z}^{\text{'}}$