Long games just beyond fixed countable length

We introduce a new type of game on natural numbers of variable countable length, which can be regarded as a diagonalization of all games of fixed countable length on natural numbers. Building on previous work by Trang and Woodin, we show that analytic determinacy of the game is equivalent to the existence of a sharp for a canonical inner model with a limit of Woodin cardinals $\lambda$ such that the order type of Woodin cardinals below $\lambda$ is $\lambda$.