#include "shortcut.h"
#include <bits/stdc++.h>
using namespace std;
#define pb push_back
#define ar array
#define all(x) x.begin(), x.end()
using i64 = long long;
const i64 inf = 1e15;
struct square{
i64 x1, x2, y1, y2;
square(i64 x1 = 0, i64 x2 = 0, i64 y1 = 0, i64 y2 = 0) : x1(x1), x2(x2), y1(y1), y2(y2) {}
square f(square q){
return square(max(x1, q.x1), min(x2, q.x2), max(y1, q.y1), min(y2, q.y2));
}
bool check(i64 x, i64 y){
return x1 <= x && x2 >= x && y1 <= y && y2 >= y;
}
};
long long find_shortcut(int n, std::vector<int> L, std::vector<int> d, int C){
vector <i64> x(n), A(n), B(n);
for ( int i = 1; i < n; i++ ){
x[i] = x[i - 1] + L[i - 1];
}
for ( int i = 0; i < n; i++ ){
A[i] = x[i] + d[i];
B[i] = x[i] - d[i];
}
vector <int> pa(n), pb(n);
for ( int i = 0; i < n; i++ ){
pa[i] = pb[i] = i;
}
sort(all(pa), [&](int &u, int &v){
return A[u] < A[v];
});
sort(all(pb), [&](int &u, int &v){
return B[u] < B[v];
});
auto ok = [&](i64 k){
i64 kk = k - C;
vector <ar<i64,3>> pt;
auto add = [&](int i, int j){
if ( i == -1 || j == -1 || i == j ) return;
if ( x[i] > x[j] ) swap(i, j);
pt.pb({x[i] + x[j], x[i] - x[j], kk - (d[i] + d[j])});
};
vector <ar<i64,2>> q(4, {inf, -1}), qq(4, {inf, -1});
auto upd = [&](int j){
vector <i64> t = {A[j], -A[j], B[j], -B[j]};
for ( int i = 0; i < 4; i++ ){
if ( q[i][0] > t[i] ){
qq[i] = q[i];
q[i] = {t[i], j};
} else if ( qq[i][0] > t[i] ){
qq[i] = {t[i], j};
}
}
};
int j = 0;
for ( auto &i: pa ){
while ( j < n && A[i] - k > B[pb[j]] ){
upd(pb[j]); j++;
}
for ( auto &[vv, j]: q ) add(i, j);
for ( auto &[v, j]: qq ) add(i, j);
}
if ( pt.empty() ) return true;
auto [X, Y, d] = pt[0];
square t = square(X - d, X + d, Y - d, Y + d);
for ( int i = 1; i < (int)pt.size(); i++ ){
auto [x, y, d] = pt[i];
t = t.f(square(x - d, x + d, y - d, y + d));
}
if ( t.x1 > t.x2 || t.y1 > t.y2 ){ // area 0
return false;
}
int l = n, r = n - 1, tl = 0, tr = -1;
for ( int i = 0; i < n; i++ ){
while ( l > 0 && t.x1 - x[i] <= x[l - 1] ) l--;
while ( r >= 0 && t.x2 - x[i] < x[r] ) r--;
while ( tr + 1 < n && x[tr + 1] <= -t.y1 + x[i] ) tr++;
while ( tl < n && x[tl] < -t.y2 + x[i] ) tl++;
int a = max(tl, l), b = min(tr, r);
if ( a <= b && (b != a || b != i) ){
return true;
}
}
return false;
};
i64 l = 0, r = 1e15;
while ( l < r ){
i64 m = (l + r) / 2;
if ( ok(m) ){
r = m;
} else l = m + 1;
}
return l;
}
