200872: [AtCoder]ARC087 E - Prefix-free Game

Problem Statement

For strings $s$ and $t$, we will say that $s$ and $t$ are prefix-free when neither is a prefix of the other.

Let $L$ be a positive integer. A set of strings $S$ is a good string set when the following conditions hold true:

• Each string in $S$ has a length between $1$ and $L$ (inclusive) and consists of the characters 0 and 1.
• Any two distinct strings in $S$ are prefix-free.

We have a good string set $S = \{ s_1, s_2, ..., s_N \}$. Alice and Bob will play a game against each other. They will alternately perform the following operation, starting from Alice:

• Add a new string to $S$. After addition, $S$ must still be a good string set.

The first player who becomes unable to perform the operation loses the game. Determine the winner of the game when both players play optimally.

Constraints

• $1 \leq N \leq 10^5$
• $1 \leq L \leq 10^{18}$
• $s_1$, $s_2$, ..., $s_N$ are all distinct.
• { $s_1$, $s_2$, ..., $s_N$ } is a good string set.
• $|s_1| + |s_2| + ... + |s_N| \leq 10^5$

Input

Input is given from Standard Input in the following format:

$N$ $L$
$s_1$
$s_2$
$:$
$s_N$

Output

If Alice will win, print Alice; if Bob will win, print Bob.

Sample Input 1

2 2
00
01

Sample Output 1

Alice

If Alice adds 1, Bob will be unable to add a new string.

Sample Input 2

2 2
00
11

Sample Output 2

Bob

There are two strings that Alice can add on the first turn: 01 and 10. In case she adds 01, if Bob add 10, she will be unable to add a new string. Also, in case she adds 10, if Bob add 01, she will be unable to add a new string.

Sample Input 3

3 3
0
10
110

Sample Output 3

Alice

If Alice adds 111, Bob will be unable to add a new string.

Sample Input 4

2 1
0
1

Sample Output 4

Bob

Alice is unable to add a new string on the first turn.

Sample Input 5

1 2
11

Sample Output 5

Alice

Sample Input 6

2 3
101
11

Sample Output 6

Bob