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2183A — Binary Array Game

First, if the entire sequence consists of $1$s, Alice wins immediately by operating on the whole sequence.

Note that the last operation must involve at least one of $$$a_1$$$ or $$$a_n$$$. We discuss based on the values of $$$a_1$$$ and $$$a_n$$$:

If $$$a_1 = 1$$$, then Alice can directly operate on the interval $$$[2,n]$$$ to win, because $$$a_{2 \sim n}$$$ must contain at least one $$$0$$$.

If $$$a_n = 1$$$, then Alice can directly operate on the interval $$$[1,n-1]$$$ to win, because $$$a_{1 \sim n-1}$$$ must contain at least one $$$0$$$.

If $$$a_1 = 0$$$ and $$$a_n = 0$$$, then operating on the entire sequence would certainly cause Alice to lose. This implies that Alice cannot operate on both $$$a_1$$$ and $$$a_n$$$ simultaneously. Therefore, after Alice's operation, there will still be at least one $$$0$$$ in the sequence $$$a$$$. Bob can then operate on the entire sequence to win. Hence, in this case, Bob wins.

Time complexity: $$$O(\sum n)$$$.