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 Rust (1.15.1) (judge verdict: Accepted)

fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();
    let l: usize = sc.read();

    let mut trie = Trie::new(0);
    for _ in 0..n {
        trie.add(sc.chars(), 0);
    }

    let mut result = vec![];
    trie.count(&mut result);

    let mut ans = 0;
    for grundy in result
        .into_iter()
        .map(|depth| (l - depth).trailing_zeros() + 1)
    {
        ans ^= grundy;
    }

    println!("{}", if ans == 0 { "Bob" } else { "Alice" });
}

#[derive(Debug)]
struct Trie {
    one: Option<Box<Trie>>,
    zero: Option<Box<Trie>>,
    depth: usize,
}

impl Trie {
    fn new(depth: usize) -> Self {
        Trie {
            zero: None,
            one: None,
            depth: depth,
        }
    }

    fn add(&mut self, s: Vec<char>, i: usize) {
        if s.len() <= i {
            return;
        }
        match s[i] {
            '0' => {
                if self.zero.is_none() {
                    self.zero = Some(Box::new(Trie::new(self.depth + 1)));
                }
                self.zero.as_mut().unwrap().add(s, i + 1);
            }
            '1' => {
                if self.one.is_none() {
                    self.one = Some(Box::new(Trie::new(self.depth + 1)));
                }
                self.one.as_mut().unwrap().add(s, i + 1);
            }
            _ => unreachable!(),
        }
    }

    fn count(&self, result: &mut Vec<usize>) {
        let mut c = 0;
        match self.one {
            Some(ref one) => one.count(result),
            None => c += 1,
        }
        match self.zero {
            Some(ref zero) => zero.count(result),
            None => c += 1,
        }
        if c == 1 {
            result.push(self.depth);
        }
    }
}

pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n')
            .take_while(|&b| b != b' ' && b != b'\n')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}