A. Everything Nim
games; greedy; math; sortings
Alice and Bob are playing a game on \(n\) piles of stones. On each player's turn, they select a positive integer \(k\) that is at most the size of the smallest nonempty pile and remove \(k\) stones from each nonempty pile at once. The first player who is unable to make a move (because all piles are empty) loses.Given that Alice goes first, who will win the game if both players play optimally?
The first line of the input contains a single integer \(t\) (\(1 \le t \le 10^4\)) — the number of test cases. The description of the test cases follows.The first line of each test case contains a single integer \(n\) (\(1 \le n \le 2\cdot 10^5\)) — the number of piles in the game.The next line of each test case contains \(n\) integers \(a_1, a_2, \ldots a_n\) (\(1 \le a_i \le 10^9\)), where \(a_i\) is the initial number of stones in the \(i\)-th pile.It is guaranteed that the sum of \(n\) over all test cases does not exceed \(2\cdot 10^5\).
For each test case, print a single line with the name of the winner, assuming both players play optimally. If Alice wins, print ""Alice"", otherwise print ""Bob"" (without quotes).
In the first test case, Alice can win by choosing \(k=3\) on her first turn, which will empty all of the piles at once.In the second test case, Alice must choose \(k=1\) on her first turn since there is a pile of size \(1\), so Bob can win on the next turn by choosing \(k=6\).