9122: Non-Puzzle: Game

Alice and Bob are playing a game.

There are $n$ numbers on the black board, where the $i$-th number is aia_{i}ai. Alice and Bob play in turns. Alice plays first. The current player can choose two numbers aia_{i}ai and aja_{j}aj from the black board ($i$ and $j$ can be equal), and write a new number ai⊕aja_{i} \oplus a_{j}ai⊕aj on the blackboard, where $\oplus$ denotes bitwise-xor operation. The players cannot skip their turns. The player who writes number $k$ on the blackboard wins.

If no one writes $k$ on the blackboard in 1010010^{100}10100 rounds, then the game ends in a draw.

Now, assume Alice and Bob both play optimally, determine the outcome of the game (Alice wins, Bob wins, or draw).

There are multiple test cases.
The first line contains one integer $T$ (1≤T≤5⋅1041\le T \le 5 \cdot 10^41≤T≤5⋅104), representing the number of test cases.
For each test case, the first line contains two integers n,kn,kn,k (1≤n≤1061\le n\le 10^61≤n≤106,0≤k<2300 \le k < 2^{30}0≤k<230).
The following line contains $n$ integers aia_{i}ai (0≤ai<2300 \le a_{i} < 2^{30}0≤ai<230).
It is guaranteed that the sum of $n$ does not exceed 10610^6106.

For each test case, output one line containing "$\text{Alice}$", "$\text{Bob}$" or "$\text{Draw}$", representing the result of the game.

5
5 4
1 7 3 2 5
2 0
1 1
5 3
1 5 7 9 5
4 2
6 4 3 1
1 1
1

Alice
Alice
Draw
Alice
Bob