Swap Game
Description
Alice and Bob are playing a game on an array $a$ of $n$ positive integers. Alice and Bob make alternating moves with Alice going first.
In his/her turn, the player makes the following move:
- If $a_1 = 0$, the player loses the game, otherwise:
- Player chooses some $i$ with $2\le i \le n$. Then player decreases the value of $a_1$ by $1$ and swaps $a_1$ with $a_i$.
Determine the winner of the game if both players play optimally.
The input consists of multiple test cases. The first line contains a single integer $t$ $(1 \leq t \leq 2 \cdot 10^4)$ — the number of test cases. The description of the test cases follows.
The first line of each test case contains a single integer $n$ $(2 \leq n \leq 10^5)$ — the length of the array $a$.
The second line of each test case contains $n$ integers $a_1,a_2 \ldots a_n$ $(1 \leq a_i \leq 10^9)$ — the elements of the array $a$.
It is guaranteed that sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
For each test case, if Alice will win the game, output "Alice". Otherwise, output "Bob".
You can output each letter in any case. For example, "alIcE", "Alice", "alice" will all be considered identical.
题面翻译
Alice 和 Bob 两个人在玩游戏。
有一个长度为 的序列 ，Alice 和 Bob 两人轮流完成一个操作，Alice 先开始。
每个人可以将数列的第一个数减 ，并将它与后面序列的一个数进行交换，如果一个人操作之前发现当前序列中的第一个数为 ，这个人就输了。
问如果两人都足够聪明，最后谁会赢？
Input
The input consists of multiple test cases. The first line contains a single integer $t$ $(1 \leq t \leq 2 \cdot 10^4)$ — the number of test cases. The description of the test cases follows.
The first line of each test case contains a single integer $n$ $(2 \leq n \leq 10^5)$ — the length of the array $a$.
The second line of each test case contains $n$ integers $a_1,a_2 \ldots a_n$ $(1 \leq a_i \leq 10^9)$ — the elements of the array $a$.
It is guaranteed that sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case, if Alice will win the game, output "Alice". Otherwise, output "Bob".
You can output each letter in any case. For example, "alIcE", "Alice", "alice" will all be considered identical.
3
2
1 1
2
2 1
3
5 4 4
Bob
Alice
Alice
Note
In the first testcase, in her turn, Alice can only choose $i = 2$, making the array equal $[1, 0]$. Then Bob, in his turn, will also choose $i = 2$ and make the array equal $[0, 0]$. As $a_1 = 0$, Alice loses.
In the second testcase, once again, players can only choose $i = 2$. Then the array will change as follows: $[2, 1] \to [1, 1] \to [1, 0] \to [0, 0]$, and Bob loses.
In the third testcase, we can show that Alice has a winning strategy.
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