def solve():
n = int(input())
x = [int(i) for i in input().split()]
t = [int(i) for i in input().split()]
l, r = 0, 1e10
pos1, pos2 = 0, 1e8
for i in range(n):
l = max(l, t[i])
pos1 = min(pos1, x[i])
pos2 = max(pos2, x[i])
eps = 1e-8
m = 0
while(r > l + eps):
m = (l + r) / 2
p1,p2 = pos1, pos2
for i in range(n):
p1 = max(p1, x[i] - (m - t[i]))
p2 = min(p2, x[i] + (m + t[i]))
if p1 > p2:
l = m
else:
r = m
for i in range(n):
pos1 = max(pos1, x[i] - (m - t[i]))
pos2 = min(pos2, x[i] + (m - t[i]))
x = (pos1 + pos2) / 2
print('%.7f' % x)
T = int(input())
while(T):
solve()
T -= 1#include <bits/stdc++.h>
using namespace std;
#define fore(i, l, r) for (lli i = (l) - ((l) > (r)); i != (r) - ((l) > (r)); i += 1 - 2 * ((l) > (r)))
#define all(a) begin(a), end(a)
#define sz(a) lli(a.size())
#define pb push_back
#define f first
#define s second
#ifdef LOCAL
#include "../debug.h"
#else
#define debug(...)
#endif
using lli = long long;
using ld = long double;
using ii = pair<lli, lli>;
const lli N = 1e6 + 5;
const ld INF = 4e18 + 5;
const ld EPS = 1e-20;
#define eq(a, b) (abs((a) - (b)) <= +EPS)
#define neq(a, b) (!eq(a, b))
#define geq(a, b) ((a) - (b) >= -EPS)
#define leq(a, b) ((a) - (b) <= +EPS)
#define ge(a, b) ((a) - (b) > +EPS)
#define le(a, b) ((a) - (b) < -EPS)
lli n;
ii a[N];
template <class T>
bool umin(T& a, T b) {
return (a = (le(a, b) ? a : b)) == b;
}
template <class T>
bool umax(T& a, T b) {
return (a = (ge(a, b) ? a : b)) == b;
}
template <class T, class F>
pair<T, T> ternary(T lo, T hi, F f) {
lli original_lo = lo;
lli original_hi = hi;
fore (it, 0, 234) {
T m1 = lo + (hi - lo) / 3.0;
T m2 = hi - (hi - lo) / 3.0;
if (leq(f(m1), f(m2)))
hi = m2;
else
lo = m1;
}
return min({make_pair(f(lo), lo), make_pair(f(hi), hi)});
}
void testCase() {
cin >> n;
multiset<lli> left, right;
fore (i, 0, n) {
cin >> a[i].f;
}
fore (i, 0, n) {
cin >> a[i].s;
right.insert(a[i].f + a[i].s);
}
ld ans = 0;
ld mx = INF;
ld prev = 0;
ld delta = 0;
sort(a, a + n);
fore (i, 0, n) {
delta += a[i].f - prev;
prev = a[i].f;
right.erase(right.find(a[i].f + a[i].s));
left.insert(a[i].s - delta);
lli mx_left = left.empty() ? 0 : *left.rbegin() + delta;
lli mx_right = right.empty() ? 0 : *right.rbegin() - delta;
if (umin<ld>(mx, max(mx_left, mx_right))) {
ans = a[i].f;
}
if (i < n - 1 && a[i].f != a[i + 1].f) {
debug(a[i], a[i + 1]);
auto [best, x] = ternary<ld>(a[i].f, a[i + 1].f, [&](ld x) {
ld cur_mx_left = mx_left + (x - a[i].f);
ld cur_mx_right = mx_right - (x - a[i].f);
ld mx = cur_mx_left;
umax(mx, cur_mx_right);
return mx;
});
if (umin(mx, best)) {
debug(mx, best);
ans = x;
}
}
}
cout << fixed << setprecision(10) << ans << '\n';
}
int main() {
cin.tie(0)->sync_with_stdio(0), cout.tie(0);
lli tc = 1;
cin >> tc;
while (tc--) {
testCase();
}
return 0;
}
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