Glass Bead Game

You play the following game against a devilishly clever opponent. On the table, there are two bowls filled with glass beads. In each move, depending on whose turn it is, you or your opponent can do one of the following. - Move one bead from the first bowl to the second bowl. - Remove one bead from the second bowl. At any point in the game, let $x$ denote the number of beads in the first bowl and $y$