Mixi Game

Vanja and Miksi have already finished everything for this year at their university, so they decided to spend their free time playing a game with a *binary* sequence $A_1, A_2, \dots, A_N$ (i.e. a sequence containing only integers $0$ and $1$). At the beginning of the game, Vanja and Miksi write down the expression $* A_1 * A_2 * A_3 * \ldots * A_N$ (note below that each $*$ is a placeholder and d