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++14 (GCC 5.4.1) (judge verdict: Runtime Error)

#include <bits/stdc++.h>
using namespace std;
#define int long long

int dp[1234567];

template<typename T>
void chmax(T &a,T b){
  if(a<b)a=b;
}
vector<string> s;
int ans;
int is_aaasame(string u,string t){
  if(u==t)return 2;
  int a=min((int)u.size(),(int)t.size());
  return u.substr(0,a)==t.substr(0,a);
}
int g(int a){
  if(a<1)return 0;
  if(~dp[a])return dp[a];
  set<int> se;
  int now=0;
  for(int i=a-1;i>0;i--){
    now^=g(i);
    se.insert(now);
  }
  int res=1;
  while(se.count(res))res++;
  return dp[a]=res;
}

signed main(){
  int n,l;cin>>n>>l;
  memset(dp,-1,sizeof(dp));
  s.resize(n);
  for(int i=0;i<n;i++)cin>>s[i];
  sort(s.begin(),s.end());
  queue<string> que;
  que.push("0");
  que.push("1");
  while(que.size()){
    string p=que.front();que.pop();
    int le=-1,ri=n;
    while(ri-le>1){
      int m=(ri+le)>>1;
      if(s[m]>p)ri=m;
      else le=m;
    }
    int A=-1;
    if(le>=0)chmax(A,is_aaasame(s[le],p));
    if(ri<n)chmax(A,is_aaasame(s[ri],p));
    if(A==2)continue;
    if(A==1&&p.size()<l){
      que.push(p+"0");
      que.push(p+"1");
    }
    if(!A)ans^=g(l+1-(int)p.size());
  }
  cout<<(ans?"Alice":"Bob")<<endl;
}