Each of $1000$ boxes in a line contains a single red marble, and for $1\le k\le 1000$, the box in the $k$th position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n)<\frac{1}{1000^2}$?