Monocarp is planning to host a martial arts tournament. There will be three divisions based on weight: lightweight, middleweight and heavyweight. The winner of each division will be determined by a single elimination system.

In particular, that implies that the number of participants in each division should be a power of two. Additionally, each division should have a non-zero amount of participants.

$n$ participants have registered for the tournament so far, the $i$-th of them weighs $a_i$. To split participants into divisions, Monocarp is going to establish two integer weight boundaries $x$ and $y$ ($x < y$).

All participants who weigh strictly less than $x$ will be considered lightweight. All participants who weigh greater or equal to $y$ will be considered heavyweight. The remaining participants will be considered middleweight.

It's possible that the distribution doesn't make the number of participants in each division a power of two. It can also lead to empty divisions. To fix the issues, Monocarp can invite an arbitrary number of participants to each division.

Note that Monocarp can't kick out any of the $n$ participants who have already registered for the tournament.

However, he wants to invite as little extra participants as possible. Help Monocarp to choose $x$ and $y$ in such a way that the total amount of extra participants required is as small as possible. Output that amount.


-----Input-----

The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of testcases.

The first line of each testcase contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of the registered participants.

The second line of each testcase contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le n$) — the weights of the registered participants.

The sum of $n$ over all testcases doesn't exceed $2 \cdot 10^5$.


-----Output-----

For each testcase, print a single integer — the smallest number of extra participants Monocarp is required to invite after he chooses the weight boundaries $x$ and $y$.


-----Examples-----

Input
4
4
3 1 2 1
1
1
6
2 2 2 1 1 1
8
6 3 6 3 6 3 6 6
Output
0
2
3
2


-----Note-----

In the first testcase of the example, Monocarp can choose $x=2$ and $y=3$. Lightweight, middleweight and heavyweight divisions will have $2$, $1$ and $1$ participants, respectively. They all are powers of two, so no extra participants are required.

In the second testcase of the example, regardless of the choice of $x$ and $y$, one division will have $1$ participant, the rest will have $0$. Thus, Monocarp will have to invite $1$ participant into both of the remaining divisions.

In the third testcase of the example, Monocarp can choose $x=1$ and $y=2$. Lightweight, middleweight and heavyweight divisions will have $0$, $3$ and $3$ participants, respectively. So an extra participant is needed in each division.

In the fourth testcase of the example, Monocarp can choose $x=8$ and $y=9$. Lightweight, middleweight and heavyweight divisions will have $8$, $0$ and $0$ participants, respectively. Middleweight and heavyweight division need an extra participant each.

Difficulty: HARD
Tags: Complete search

Sorting

Mathematics

Greedy algorithms
Skill types: Complete search

Sorting

Greedy algorithms

Starter code:

Solutions:
import math

class Solution:

	def MAT(self, arr):
		n = len(arr)
		d = int(math.log2(n - 1) + 1)
		arr.sort()
		cw = [1]
		for i in range(1, n):
			if arr[i] == arr[i - 1]:
				cw[-1] += 1
			else:
				cw.append(1)
		sl = [0] * (d + 1)
		c = 0
		s = 0
		for i in cw:
			s += i
			while s > 1 << c:
				c += 1
			sl[c] = s
		sr = [0] * (d + 1)
		c = 0
		s = 0
		for i in cw[-1::-1]:
			s += i
			while s > 1 << c:
				c += 1
			sr[c] = s
		ans = (1 << d) - n + 2
		for a in range(d):
			for b in range(d):
				lr = sl[a] + sr[b]
				c = 1
				while c < n - lr:
					c <<= 1
				ans = min(ans, (1 << a) + (1 << b) + c - n)
		return ans
t = int(input())
ans = Solution()
for i in range(t):
	n = int(input())
	arr = [int(x) for x in input().split()]
	if n == 1:
		print(2)
	else:
		print(ans.MAT(arr))