A Rectangle on a Parabola

In the coordinate plane, a point $P$ satisfies: the distance from $P$ to the $x$-axis equals the distance from $P$ to the point $\left(0, \dfrac{1}{2}\right)$. Let $W$ be the locus of $P$.

1. Find the equation of $W$. 2. A rectangle $ABCD$ has three of its vertices on $W$. Prove that the perimeter of rectangle $ABCD$ is greater than $3\sqrt{3}$.