For $n\ge 1$, let $d_n$ denote the length of the line segment connecting the two points where the line $y = x + n + 1$ intersects the parabola $8x^2 = y - \frac{1}{32}$ . Compute the sum $$\sum_{n=1}^{1000}\frac{1}{n\cdot d_n^2}$$