Two people take turns writing natural numbers from $1$ to $1000$. On the first move, the first player writes the number $1$ on the board. Then with your next move you can write either the number $2a$ or the number $a+1$ on the board if number $a$ is already written on the board. In this case, it is forbidden to write down numbers that are already written on the board. The one who writes out wins the number $1000$ on the board. Who wins if played correctly?