Chef and Deque

Chef and one of his friends were practicing for ICPC together by giving each other programming problems. In Chef's turn, Chef gave his friend a modified deque (double-ended queue), initially with $N$ elements $A_1, A_2, \ldots, A_N$, which supported performing operations of the following type: - Choose an integer $K$, which must be a power of $2$ (possibly $2^0 = 1$). Additionally, each particula