A point is chosen at random within the square in the coordinate plane whose vertices are $(0,0)$, $(1000,0)$, $(1000,1000)$, and $(0,1000)$. The probability that the point is within $d$ units of a lattice point is $\frac{1}{4}$. (A point $(x,y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest hundredth?