TUYMAADA 2023 J2

Serge and Tanya want to show Masha a magic trick. Serge leaves the room. Masha writes down a sequence $(a_1, a_2, \ldots , a_n)$ , where all $a_k$ equal $0$ or $1$ . After that Tanya writes down a sequence $(b_1, b_2, \ldots , b_n)$ , where all $b_k$ also equal $0$ or $1$ . Then Masha either does nothing or says "Mutabor" and replaces both sequences: her own sequence by $(a_n, a_{n-1}, \ldots , a_1)$ , and Tanya's sequence by $(1 - b_n, 1 - b_{n-1}, \ldots , 1 - b_1)$ . Masha's sequence is covered by a napkin, and Serge is invited to the room. Serge should look at Tanya's sequence and tell the sequence covered by the napkin. For what $n$ Serge and Tanya can prepare and show such a trick? Serge does not have to determine whether the word "Mutabor" has been pronounced.