Stone Picking Game

Problem Description

There is a pile of $x$ stones. Alice and Bob take turns making moves, with Alice going first. In each of her turns, Alice may take $a_1, a_2, \dots$, or $a_{n_1}$ stones (that is, choose some $a_i$ and remove exactly $a_i$ stones); in each of his turns, Bob may take $b_1, b_2, \dots$, or $b_{n_2}$ stones. Whoever cannot make a move loses.

For every $x = 1, 2, \dots, m$, determine who wins.

Input Format

The first line contains three integers $n_1, n_2, m$ ($n_1, n_2 \le 2000$, $m \le 2000$).

The second line contains $n_1$ integers $a_1, a_2, \dots, a_{n_1}$ ($1 \le a_i \le m$).

The third line contains $n_2$ integers $b_1, b_2, \dots, b_{n_2}$ ($1 \le b_i \le m$).

Output Format

Output $m$ lines, each containing either Alice or Bob, indicating the winner for that value of $x$.

Sample Input

3 5 10
1 3 4
2 3 5 6 7

Sample Output

Alice
Alice
Alice
Alice
Alice
Bob
Alice
Alice
Bob
Bob

Constraints

$n_1, n_2 \le 2000$, $m \le 2000$, $1 \le a_i, b_i \le m$.