CF2160A MEX Partition

Let a partition of a multiset $ B $ be a collection of multisets $ s_1, s_2,\ldots, s_k $ such that each element appears the same number of times in $ B $ and across all of $ s_1,s_2,\ldots,s_k $ . For example, some partitions of $ \{1,2,3,3\} $ include $ \{1,3\}+\{2,3\}, \{1,2,3,3\} $ , and $ \{2\}+\{1,3\}+\{3\} $ , but not $ \{1,2\}+\{3\} $ . A partition is called valid if the $ \operatorname{mex} $ $ ^{\text{∗}} $ of all multisets in the partition is the same. The score of a valid partition is the $ \operatorname{mex} $ of any multiset in the partition. You are given a multiset $ A $ of size $ n $ . Find the minimum score over all valid partitions of $ A $ . $ ^{\text{∗}} $ The minimum excluded (MEX) of a collection of integers $ c_1, c_2, \ldots, c_k $ is defined as the smallest non-negative integer $ x $ which does not occur in the collection $ c $ .

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 100 $ ). The description of the test cases follows. The first line of each test case contains an integer $ n $ ( $ 1 \leq n \leq 100 $ ). The second line contains $ n $ integers $ A_1, A_2, \ldots, A_n $ denoting the elements of $ A $ ( $ 0 \leq A_i \leq 100 $ ). It is not guaranteed that the elements are given in non-decreasing order.

For each test case, output the minimum score over all valid partitions.

In the first test case, the minimum score of $ 1 $ can be obtained by the partition $ \{0\}+\{0\}+\{0\} $ . The partition is valid because each multiset has a $ \operatorname{mex} $ of $ 1 $ , which is consequently the score of the partition. In the second test case, we can use $ \{1,2\} $ as the only multiset in the partition, which has $ \operatorname{mex} 0 $ .