B'This is the hard version of the problem. The only difference is that in this version everyone must give candies to no more than one person and receive candies from no more than one person. Note that a submission cannot pass both versions of the problem at the same time. You can make hacks only if both versions of the problem are solved. After Zhongkao examination, Daniel and his friends are going to have a party. Everyone will come with some candies. There will be n people at the party. Initially, the i -th person has a_i candies. During the party, they will swap their candies. To do this, they will line up in an arbitrary order and everyone will do the following no more than once: Daniel likes fairness, so he will be happy if and only if everyone receives candies from no more than one person. Meanwhile, his friend Tom likes average, so he will be happy if and only if all the people have the same number of candies after all swaps. Determine whether there exists a way to swap candies, so that both Daniel and Tom will be happy after the swaps. The first line of input contains a single integer t ( 1 <= t <= 1000 ) -- the number of test cases. The description of test cases follows. The first line of each test case contains a single integer n ( 2 <= n <= 2 cdot 10^5 ) -- the number of people at the party. The second line of each test case contains n integers a_1,a_2, ldots,a_n ( 1 <= a_i <= 10^9 ) -- the number of candies each person has. It is guaranteed that the sum of n over all test cases does not exceed 2 cdot 10^5 . For each test case, print "Yes" (without quotes) if exists a way to swap candies to make both Daniel and Tom happy, and print "No" (without quotes) otherwise. You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses. In the first test case, the second person gives 1 candy to th'...