You are given an $n \times n$ matrix filled with the numbers $1, 2, 3, \ldots, n^2$. The matrix is arranged such that every row (from left to right) and every column (from top to bottom) is sorted in strictly increasing order. Let $a_{ij}$ denote the number at the intersection of the $i$-th row and $j$-th column.

For each $i$ where $1 \leq i \leq n$, define $b_i$ as the number of distinct entries that can appear in the diagonal position $a_{ii}$. Your task is to find the sum $b_1 + b_2 + \cdots + b_n$ for $n = 1000$.