#include "shortcut.h"
#include <bits/stdc++.h>
#define N 1000005
using namespace std;
long long p[N];
long long d[N];
int n,nwedge;
vector<pair<long long,int>> a;
vector<pair<long long,pair<int,int>>> b;
long long val1[N],val2[N];
bool ck(long long k){
for(int i = 0;i<n;i++){
val1[i] = -1e18;
val2[i] = 1e18;
}
int pt = n;
for(int i = 0;i<n;i++){
while(pt && b[pt-1].first + a[i].first > k){
pt--;
}
if(pt != n){
val1[(b[pt].second.first == a[i].second?b[pt].second.second:b[pt].second.first)] = max(val1[(b[pt].second.first == a[i].second?b[pt].second.second:b[pt].second.first)],p[a[i].second] + d[a[i].second]);
val2[(b[pt].second.first == a[i].second?b[pt].second.second:b[pt].second.first)] = min(val2[(b[pt].second.first == a[i].second?b[pt].second.second:b[pt].second.first)],p[a[i].second] - d[a[i].second]);
}
}
long long xl = -1e18, xr = 1e18;
long long yl = -1e18, yr = 1e18;
stack<int> st;
for(int i = 0;i<n;i++){
while(st.size() && p[st.top()] + d[st.top()] <= p[i] + d[i]){
val1[i] = max(val1[i],val1[st.top()]);
val2[i] = min(val2[i],val2[st.top()]);
st.pop();
}
xl = max(xl,+ p[i] - (k - d[i] - nwedge) + val1[i]);
xr = min(xr,+ p[i] + (k - d[i] - nwedge) + val2[i]);
yl = max(yl,- p[i] - (k - d[i] - nwedge) + val1[i]);
yr = min(yr,- p[i] + (k - d[i] - nwedge) + val2[i]);
st.push(i);
}
if(xl > xr || yl > yr)return 0;
int pt1 = n,pt2 = n-1;
int pt3 = -1,pt4 = 0;
for(int i = 0;i<n;i++){
while(pt1 && p[pt1-1] + p[i] >= xl){
pt1--;
}
while(pt2 && p[pt2] + p[i] > xr){
pt2--;
}
while(pt3 + 1 != n && -p[pt3+1] + p[i] >= yl){
pt3++;
}
while(pt4 != n && -p[pt4] + p[i] > yr){
pt4++;
}
int l1 = pt1,r1 = pt2;
int l2 = pt4,r2 = pt3;
if(r1 < l1 || r2 < l2)continue;
if(r1 < l2 || r2 < l1)continue;
return 1;
}
return 0;
}
long long find_shortcut(int n_, vector<int> l_, vector<int> d_, int nwedge_)
{
n = n_;
nwedge = nwedge_;
for(int i = 1;i<n;i++){
p[i] = p[i-1] + l_[i-1];
}
for(int i = 0;i<n;i++){
d[i] = d_[i];
}
for(int i = 0;i<n;i++){
a.push_back({-p[i] + d[i],i});
}
for(int i = 0;i<n;i++){
b.push_back({p[i] + d[i],{i,n}});
}
sort(a.begin(),a.end());
sort(b.begin(),b.end());
for(int i = b.size()-2;i>=0;i--){
if(min(b[i].second.first,b[i+1].second.first) == b[i].second.first){
b[i].second.second = min(b[i].second.second,b[i+1].second.first);
}
else b[i].second.second = min(b[i].second.first,b[i+1].second.second);
b[i].second.first = min(b[i].second.first,b[i+1].second.first);
}
long long l = 0, r = 1e16;
while(l < r){
long long m = (l + r)/2;
if(ck(m)){
r = m;
}
else l = m+1;
}
return l;
}
