$n$ people live on the coordinate line, the $i$-th one lives at the point $x_i$ ($1 \le i \le n$). They want to choose a position $x_0$ to meet. The $i$-th person will spend $|x_i - x_0|$ minutes to get to the meeting place. Also, the $i$-th person needs $t_i$ minutes to get dressed, so in total he or she needs $t_i + |x_i - x_0|$ minutes.

Here $|y|$ denotes the absolute value of $y$.

These people ask you to find a position $x_0$ that minimizes the time in which all $n$ people can gather at the meeting place.


-----Input-----

The first line contains a single integer $t$ ($1 \le t \le 10^3$) — the number of test cases. Then the test cases follow.

Each test case consists of three lines.

The first line contains a single integer $n$ ($1 \le n \le 10^5$) — the number of people.

The second line contains $n$ integers $x_1, x_2, \dots, x_n$ ($0 \le x_i \le 10^{8}$) — the positions of the people.

The third line contains $n$ integers $t_1, t_2, \dots, t_n$ ($0 \le t_i \le 10^{8}$), where $t_i$ is the time $i$-th person needs to get dressed.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.


-----Output-----

For each test case, print a single real number — the optimum position $x_0$. It can be shown that the optimal position $x_0$ is unique.

Your answer will be considered correct if its absolute or relative error does not exceed $10^{−6}$. Formally, let your answer be $a$, the jury's answer be $b$. Your answer will be considered correct if $\frac{|a−b|}{max(1,|b|)} \le 10^{−6}$.


-----Examples-----

Input
7
1
0
3
2
3 1
0 0
2
1 4
0 0
3
1 2 3
0 0 0
3
1 2 3
4 1 2
3
3 3 3
5 3 3
6
5 4 7 2 10 4
3 2 5 1 4 6
Output
0
2
2.5
2
1
3
6


-----Note-----

In the $1$-st test case there is one person, so it is efficient to choose his or her position for the meeting place. Then he or she will get to it in $3$ minutes, that he or she need to get dressed.

In the $2$-nd test case there are $2$ people who don't need time to get dressed. Each of them needs one minute to get to position $2$.

In the $5$-th test case the $1$-st person needs $4$ minutes to get to position $1$ ($4$ minutes to get dressed and $0$ minutes on the way); the $2$-nd person needs $2$ minutes to get to position $1$ ($1$ minute to get dressed and $1$ minute on the way); the $3$-rd person needs $4$ minutes to get to position $1$ ($2$ minutes to get dressed and $2$ minutes on the way).

Difficulty: MEDIUM_HARD

Solutions:
t = int(input())
for _ in range(t):
	n = int(input())
	x = list(map(int, input().split(' ')))
	t = list(map(int, input().split(' ')))
	r = [a + b for (a, b) in zip(x, t)]
	l = [a - b for (a, b) in zip(x, t)]
	o = (max(r) + min(l)) / 2
	print(int(o) if o % 1 == 0 else o)


t = int(input())

def solve():
	n = int(input())
	lst = list(map(int, input().split()))
	t = list(map(int, input().split()))
	arr = []
	for i in range(n):
		arr += [lst[i] - t[i], lst[i] + t[i]]
	s = max(arr) + min(arr)
	if s % 2 == 0:
		print(s // 2)
	else:
		print(s // 2, end='')
		print('.', end='')
		print(5)
while t:
	solve()
	t -= 1


for _ in range(int(input())):
	a = int(input())
	b = [int(i) for i in input().split()]
	c = [int(i) for i in input().split()]
	d = [c[i] + b[i] for i in range(a)]
	e = [b[i] - c[i] for i in range(a)]
	d.extend(e)
	(mi, ma) = (min(d), max(d))
	if (mi + ma) % 2 == 0:
		print((mi + ma) // 2)
	else:
		print((mi + ma) / 2)


for _ in range(int(input())):
	n = int(input())
	x = list(map(int, input().split()))
	t = list(map(int, input().split()))
	arr = []
	for i in range(n):
		arr.append(x[i] - t[i])
		arr.append(x[i] + t[i])
	ans = (max(arr) + min(arr)) / 2
	if int(ans) == ans:
		print(int(ans))
	else:
		print(ans)


s = int(input())
while s != 0:
	n = int(input())
	x = list(map(int, input().split()))
	t = list(map(int, input().split()))
	mx = x[0] + t[0]
	mi = x[0] - t[0]
	for i in range(1, n):
		mi = min(mi, x[i] - t[i])
		mx = max(mx, x[i] + t[i])
	sum = mi + mx
	if sum % 2 == 0:
		print(sum // 2)
	else:
		print(str(sum // 2) + '.5')
	s -= 1


from math import fabs
t = int(input())
for i in range(t):
	n = int(input())
	line1 = input().split(' ')
	line2 = input().split(' ')
	intline = [int(x) for x in line1]
	intline2 = [int(x) for x in line2]
	x0 = intline[0]
	s0 = intline2[0]
	for j in range(1, len(intline)):
		x1 = intline[j]
		s1 = intline2[j]
		if x0 > x1:
			(x0, x1) = (x1, x0)
			(s0, s1) = (s1, s0)
		if x1 - x0 <= fabs(s1 - s0):
			if s1 > s0:
				x0 = x1
				s0 = s1
			else:
				pass
		else:
			x0 = (x1 + x0 + (s1 - s0)) / 2
			s0 = s1 + (x1 - x0)
	if x0 % 1 == 0.0:
		print(int(x0))
	else:
		print(x0)


for _ in range(int(input())):
	n = int(input())
	arr = input().split(' ')
	time = input().split(' ')
	arr = [int(i) for i in arr]
	time = [int(i) for i in time]
	(a, b) = (-2 ** 31, -2 ** 31)
	for i in range(n):
		a = max(a, time[i] + arr[i])
		b = max(b, time[i] - arr[i])
	if (a - b) % 2:
		print((a - b) / 2)
	else:
		print((a - b) // 2)


nb_test = int(input())
ans = []
for _ in range(nb_test):
	n = int(input())
	xs = list(map(int, input().split(' ')))
	ts = list(map(int, input().split(' ')))
	best = 0
	worse_p = 0
	worse_n = 0

	def dist_p(i):
		return xs[i] + ts[i]

	def dist_n(i):
		return -xs[i] + ts[i]
	for i in range(n):
		if dist_p(i) >= dist_p(worse_p):
			worse_p = i
		if dist_n(i) >= dist_n(worse_n):
			worse_n = i
	an = (xs[worse_p] + xs[worse_n] + ts[worse_p] - ts[worse_n]) / 2
	if an.is_integer():
		an = int(an)
	ans.append(an)
for an in ans:
	print(an)


ntestcases = int(input())
for t in range(ntestcases):
	numPeople = int(input())
	positions = list(map(int, input().split(' ')))
	dressingTimes = list(map(int, input().split(' ')))
	if dressingTimes[0] > 0:
		minPos = positions[0] - dressingTimes[0]
		maxPos = positions[0] + dressingTimes[0]
	else:
		minPos = positions[0]
		maxPos = positions[0]
	for i in range(1, numPeople):
		if dressingTimes[i] > 0:
			smol = positions[i] - dressingTimes[i]
			big = positions[i] + dressingTimes[i]
			minPos = min(minPos, smol)
			maxPos = max(maxPos, big)
		else:
			minPos = min(minPos, positions[i])
			maxPos = max(maxPos, positions[i])
	answer = (maxPos + minPos) / 2
	intAnswer = int(answer)
	if answer == intAnswer:
		print(intAnswer)
	else:
		print(answer)


t = int(input())
for _ in range(t):
	n = int(input())
	(list_time, list_cord) = ([], [])
	list_cord = list(map(int, input().split()))
	list_time = list(map(int, input().split()))
	list_all_x_points = [0 for i in range(2 * n)]
	for i in range(0, 2 * n, 2):
		list_all_x_points[i] = list_cord[i // 2] + list_time[i // 2]
		list_all_x_points[1 + i] = list_cord[i // 2] - list_time[i // 2]
	maxi = mini = list_all_x_points[0]
	for i in list_all_x_points:
		maxi = max(maxi, i)
		mini = min(mini, i)
	sum = mini + maxi
	print(sum // 2, end='')
	if sum % 2:
		print('.5')
	else:
		print()


from collections import Counter
for _ in range(int(input())):
	n = int(input())
	arr = list(map(int, input().split()))
	time = list(map(int, input().split()))
	x = []
	for i in range(n):
		x.append([arr[i], time[i]])
	x.sort()
	ans = 0
	mx = max(time)
	idx = -1
	md = 0
	for i in range(n):
		if x[i][1] == mx:
			idx = i
			md = x[i][0]
			break
	a = md
	b = md
	for i in range(n):
		if i < idx:
			x[i][0] += mx - x[i][1]
			x[i][0] = min(md, x[i][0])
			a = min(a, x[i][0])
		elif i > idx:
			x[i][0] -= mx - x[i][1]
			x[i][0] = max(md, x[i][0])
			b = max(b, x[i][0])
	ans = (b - a) / 2 + a
	if ans.is_integer():
		print(int(ans))
	else:
		print(ans)


import sys
t = int(sys.stdin.readline())
for i in range(t):
	n = int(input())
	lstx = list(map(int, input().split()))
	lsty = list(map(int, input().split()))
	low = min(lstx)
	high = max(lstx)
	l = 0
	r = 200000001
	mid = (l + r) / 2
	ans = r
	while True:
		if r < l:
			break
		mid = (l + r) // 2
		lt = -1000000000000000.0
		rt = 1000000000000000.0
		er = 0
		for j in range(n):
			left = mid - lsty[j]
			if left < 0:
				er += 1
				break
			a = lstx[j] - left
			b = lstx[j] + left
			if a > r or b < l:
				er += 1
				break
			lt = max(lt, a)
			rt = min(rt, b)
		if er > 0:
			l = mid + 1
		else:
			ans = (lt + rt) / 2
			r = mid - 1
	if int(ans) == ans:
		ans = int(ans)
	print(ans)