Written on a blackboard are $n$ nonnegative integers whose greatest common divisor is $1$. A move consists of erasing two numbers $x$ and $y$, where $x\ge y$, on the blackboard and replacing them with the numbers $x-y$ and $2y$. Determine for which original $n$-tuples of numbers on the blackboard it is possible to reach a point, after some number of moves, where $n-1$ of the numbers on the blackboard are zeroes.