408814: GYM103328 K This is a Game

Do you know the Skylinebaby game? The game comprises $$$n$$$ positive integers $$$a_1, ..., a_n$$$, and a constant $$$k > 1$$$.

Skylinebaby game is a game for two players playing alternately. Each turn the current player can choose one of the integers $$$a_i$$$ and an integer $$$t$$$ satisfying $$$a_i \geq t \geq k$$$, then replace $$$a_i$$$ with $$$\lfloor \frac{a_i}{t} \rfloor$$$. The player who can't make a move loses the game.

Alice and Bob want to play the Skylinebaby game together, but they immediately notice that the game favors the first player a lot.

Alice and Bob want to modify the game so that it can be more even. Therefore, they propose a new idea that the player who moves secondly can decide the constant $$$k$$$ before the game starts. Of course, there should be some additional limitations of this rule, or he can just set $$$k$$$ to a large number, and the first player will not be able to make a single move. After some game testing, they agree on the final version that $$$k$$$ should not exceed any of the integers $$$a_1, \ldots, a_n$$$ ($$$k$$$ still has to be greater than 1).

Now, Alice and Bob want to play the modified version of the Skylinebaby game they just developed. Alice moves first. Given $$$n$$$ and the integers $$$a_1, ..., a_n$$$, determine the winner if both players play optimally.

The first line of the input consists of an integer $$$n$$$, representing the number of positive integers in the game.

The second line of the input consists of $$$n$$$ integers $$$a_1, a_2, ..., a_n$$$, representing the positive integers in the game.

• $$$1 \leq n \leq 10^5$$$
• $$$2 \leq a_i \leq 10^{15}$$$

If Alice wins, print Alice. Otherwise, print Bob.

1
10000

Alice

3
3 5 7

Alice

2
2 2

Bob