1965A · Everything Nim

Problem: 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?

Input Format: 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 $2\cdot 10^5$.

Output Format: 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).

Note: 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 $k=6$.