#include "shortcut.h"
#include <bits/stdc++.h>
#define IO_OP std::ios::sync_with_stdio(0); std::cin.tie(0);
#define F first
#define S second
#define V vector
#define PB push_back
#define EB emplace_back
#define MP make_pair
#define SZ(v) int((v).size())
#define ALL(v) (v).begin(), (v).end()
using namespace std;
typedef long long ll;
typedef pair<int, int> pi;
typedef V<int> vi;
const int INF = 1e9 + 7;
const ll oo = 1e18;
ll find_shortcut(int n, vi l, vi d, int c) {
assert(SZ(l) == n - 1);
assert(SZ(d) == n);
V<ll> pos(n);
for(int i = 1; i < n; i++)
pos[i] = l[i - 1] + pos[i - 1];
auto ok = [&] (ll k) {
V<pair<ll, int>> a(n), b(n);
for(int i = 0; i < n; i++) {
a[i] = {pos[i] + d[i], i};
b[i] = {k + pos[i] - d[i], i};
}
ll lb_sum = -oo, rb_sum = oo;
ll lb_dif = -oo, rb_dif = oo;
auto add_condition = [&] (int i, int j) {
assert(i < j && pos[j] - pos[i] + d[i] + d[j] > k);
// abs(x - pos[i]) + abs(y - pos[j]) <= k - (d[i] + d[j] + c)
ll dd = k - (0LL + d[i] + d[j] + c);
// pos[i] + pos[j] - d <= x + y <= pos[i] + pos[j] + d
// pos[i] - pos[j] - d <= x - y <= pos[i] - pos[j] + d
lb_sum = max(lb_sum, pos[i] + pos[j] - dd);
rb_sum = min(rb_sum, pos[i] + pos[j] + dd);
lb_dif = max(lb_dif, pos[i] - pos[j] - dd);
rb_dif = min(rb_dif, pos[i] - pos[j] + dd);
};
function<void(int, int)> cdq = [&] (int l, int r) {
if(r - l == 1) return;
int m = (l + r) / 2;
cdq(l, m), cdq(m, r);
int mxj = -1, mnj = INF;
for(int i = m - 1, j = r - 1; i >= l; i--) {
while(j >= m && a[j].F > b[i].F) {
mxj = max(mxj, a[j].S);
mnj = min(mnj, a[j].S);
j--;
}
if(mxj != -1) {
add_condition(b[i].S, mnj);
add_condition(b[i].S, mxj);
}
}
// b[i] < a[j]
inplace_merge(a.begin() + l, a.begin() + m, a.begin() + r);
inplace_merge(b.begin() + l, b.begin() + m, b.begin() + r);
};
cdq(0, n);
if(lb_sum > rb_sum) return false;
if(lb_dif > rb_dif) return false;
for(int i = 0; i < n - 1; i++) {
ll lb = max(lb_sum - pos[i], pos[i] - rb_dif), rb = min(rb_sum - pos[i], pos[i] - lb_dif);
if(lb <= rb) {
int j = lower_bound(pos.begin() + i + 1, pos.end(), lb) - pos.begin();
if(j < n && pos[j] <= rb) {
assert(lb_sum <= pos[i] + pos[j] && pos[i] + pos[j] <= rb_sum);
assert(lb_dif <= pos[i] - pos[j] && pos[i] - pos[j] <= rb_dif);
return true;
}
}
}
return false;
};
ll lb = 1, rb = 1e15;
while(lb <= rb) {
ll mb = (lb + rb) / 2;
if(ok(mb)) rb = mb - 1;
else lb = mb + 1;
}
return lb;
}
