KöMaL 2022 B5264

First Player and Second Player play the following game. First Player starts and prescribes arbitrarily many (even infinitely many) terms of a binary sequence (i.e., any term is $0$ or $1$) in a way that infinitely many terms can still be determined. Then Second Player sets the value of the first digit which has not been prescribed yet. They then repeat this procedure forever by taking turns. First Player wins if the binary sequence is periodic from a certain term, otherwise, Second Player wins. Is there a winning strategy, and if yes, who has it?