CF1928B Equalize

Vasya has two hobbies — adding permutations $ ^{\dagger} $ to arrays and finding the most frequently occurring element. Recently, he found an array $ a $ and decided to find out the maximum number of elements equal to the same number in the array $ a $ that he can obtain after adding some permutation to the array $ a $ . More formally, Vasya must choose exactly one permutation $ p_1, p_2, p_3, \ldots, p_n $ of length $ n $ , and then change the elements of the array $ a $ according to the rule $ a_i := a_i + p_i $ . After that, Vasya counts how many times each number occurs in the array $ a $ and takes the maximum of these values. You need to determine the maximum value he can obtain. $ ^{\dagger} $ A permutation of length $ n $ is an array consisting of $ n $ distinct integers from $ 1 $ to $ n $ in arbitrary order. For example, $ [2,3,1,5,4] $ is a permutation, but $ [1,2,2] $ is not a permutation ( $ 2 $ appears twice in the array), and $ [1,3,4] $ is also not a permutation ( $ n=3 $ but there is $ 4 $ in the array).

Each test consists of multiple test cases. The first line contains a single integer $ t $ ( $ 1 \leq t \leq 2 \cdot 10^4 $ ) — the number of test cases. Then follows the description of the test cases. The first line of each test case contains a single integer $ n $ ( $ 1 \le n \le 2 \cdot 10^5 $ ) — the length of the array $ a $ . The second line of each test case contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 1 \le a_i \le 10^9 $ ) — the elements of the array $ a $ . It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, output a single number — the maximum number of elements equal to the same number after the operation of adding a permutation.

In the first test case, it is optimal to choose $ p = [2, 1] $ . Then after applying the operation, the array $ a $ will be $ [3, 3] $ , in which the number $ 3 $ occurs twice, so the answer is $ 2 $ . In the second test case, one of the optimal options is $ p = [2, 3, 1, 4] $ . After applying the operation, the array $ a $ will be $ [9, 4, 5, 5] $ . Since the number $ 5 $ occurs twice, the answer is $ 2 $ .