Let $a$ be $-\frac{1}{6}$ and let $b$ be $\frac{1}{2}$. Let $x$ be the sum of the reciprocals of $a$ and $b$, let $y$ be the product of the reciprocals of $a$ and $b$, and let $z$ be twice the reciprocal of $a$ divided by the reciprocal of $b$. Calculate $x+y+z$.
