A list of $10,000$ digits is formed by repeatedly writing the digits $1,2,3,$ and $4$ from left to right, beginning $1234123412\dots$. Every third digit is then erased from the list (i.e., the $3$rd, $6$th, $9$th, $\dots$ digits from the left). From the resulting list, every second digit is erased (i.e., the $2$nd, $4$th, $6$th, $\dots$ digits from the left in what remained). Finally, every third digit is erased from what remained at that point. What is the sum of the three digits that were then in the positions $1000$, $1001$, and $1002$?