Problem:
Score : 700 points

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

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

//#pragma warning disable
using System.Collections;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;
using System;
using System.Numerics;
using System.Threading.Tasks;
using static System.Math;
using static System.Console;
class Z { static void Main() => new K(); }
class K
{
	int F() => int.Parse(ReadLine());
	long FL() => long.Parse(ReadLine());
	int[] G() => ReadLine().Split(new char[] { ' ', '\t' }, StringSplitOptions.RemoveEmptyEntries).Select(int.Parse).ToArray();
	long[] GL() => ReadLine().Split(new char[] { ' ', '\t' }, StringSplitOptions.RemoveEmptyEntries).Select(long.Parse).ToArray();
	Node root = new Node();
	readonly List<long> l = new List<long>();
	long L;
	public K()
	{
		var I = GL();
		var N = I[0];
		L = I[1];
		for (var i = 0; i < N; i++)
		{
			var s = Console.ReadLine();
			var n = root;
			for (var j = 0; j < s.Length; j++)
			{
				var x = s[j] - '0';
				n = n[x] = n[x] ?? new Node(n);
			}
		}
		DFS(root);
		var g = 0L;
		foreach (var x in l) g ^= F(x);
		Console.WriteLine(g != 0 ? "Alice" : "Bob");
	}
	long F(long n)
	{
		var c = 1L;
		while (n % 2 == 0) { c <<= 1; n >>= 1; }
		return c;
	}
	void DFS(Node n)
	{
		if (n == null) return;
		if (n.Count == 1) l.Add(L - n.Depth);
		DFS(n.Left);
		DFS(n.Right);
	}
}
class Node
{
	public Node Left, Right;
	public readonly int Depth;
	public Node this[int i] { get { return i == 0 ? Left : Right; } set { if (i == 0) Left = value; else Right = value; } }
	public int Count { get { return (Left == null ? 0 : 1) + (Right == null ? 0 : 1); } }
	public Node() { }
	public Node(Node p) { Depth = p.Depth + 1; }
}