407515: GYM102821 G Game of Primes

Alice and Bob always like playing games with each other and today they found a new game about primes.

There are two positive integers $$$x$$$ and $$$y$$$ in the game, and Alice and Bob move in turn. At each turn, the current player can choose one integer and subtract it by $$$1$$$ (making $$$(x, y)$$$ to $$$(x - 1, y)$$$ or to $$$( x, y - 1)$$$). The game ends when one of following conditions is met and the winner is specified at the same time:

• When $$$x$$$ or $$$y$$$ equals to $$$K$$$: Bob wins.
• When $$$x$$$ and $$$y$$$ are both primes: Alice wins.
• When both of the previous conditions are satisfied at the same time: Bob wins.

Now $$$x$$$, $$$y$$$, $$$K$$$ and who moves first are given, can you determine who will finally win the game if they both play optimally?

The first line of input contains an integer $$$T$$$, representing the number of test cases. Then following $$$T$$$ lines and each line contains one test case.

For each test case, there are four integers $$$x$$$, $$$y$$$, $$$K$$$ and $$$w$$$ separated by exactly one space. $$$x$$$,$$$y$$$,$$$K$$$ are mentioned above. $$$w=0$$$ when Alice moves first and $$$w = 1$$$ when Bob moves first.

For each test case, you should output Case $$$x$$$: name in one line, where $$$x$$$ indicates the case number starting from 1, and name is the player who will win the game.

4
4 9 2 0
7 10 2 0
6 39 2 0
5 28 2 0

Case 1: Alice
Case 2: Alice
Case 3: Alice
Case 4: Bob

$$$1 \le T \le 100$$$

$$$2 \le x, y \le 10^6$$$

$$$2 \le K \le \min(x, y)$$$

$$$0 \le w \le 1$$$

For $$$90\%$$$ test cases: $$$\max(x, y) \le 1000$$$