Among the $100$ constants $a_1, a_2, a_3, \ldots, a_{100}$, there are $39$ equal to $-1$ and $61$ equal to $+1$. Find the sum of all the products $a_i a_j$, where $1 \le i < j \le 100$.

Let's think step by step and output the final answer within \boxed{}.
user