Balancing Game

Two of your friends (let's denote them by $P$ and $Q$) are playing a game and you are acting as their dealer. The rules of the game are as follows: - There are $N$ bracket sequences, denoted by $S_1, S_2, \ldots, S_N$. - Initially, both players do not have any sequences. They play up to $N$ turns. For each valid $i$, in the $i$-th turn, you have to choose one player and give the bracket sequence