/*
ID: varunra2
LANG: C++
TASK: shortcut
*/
#include <bits/stdc++.h>
#include "shortcut.h"
using namespace std;
#ifdef DEBUG
#include "lib/debug.h"
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#define debug_arr(...) \
cerr << "[" << #__VA_ARGS__ << "]:", debug_arr(__VA_ARGS__)
#pragma GCC diagnostic ignored "-Wsign-compare"
//#pragma GCC diagnostic ignored "-Wunused-parameter"
//#pragma GCC diagnostic ignored "-Wunused-variable"
#else
#define debug(...) 42
#endif
#define int long long
#define EPS 1e-9
#define IN(A, B, C) assert(B <= A && A <= C)
#define INF (int)1e16
#define MEM(a, b) memset(a, (b), sizeof(a))
#define MOD 1000000007
#define MP make_pair
#define PB push_back
#define all(cont) cont.begin(), cont.end()
#define rall(cont) cont.end(), cont.begin()
#define x first
#define y second
const double PI = acos(-1.0);
typedef long long ll;
typedef long double ld;
typedef pair<int, int> PII;
typedef map<int, int> MPII;
typedef multiset<int> MSETI;
typedef set<int> SETI;
typedef set<string> SETS;
typedef vector<int> VI;
typedef vector<PII> VII;
typedef vector<VI> VVI;
typedef vector<string> VS;
#define rep(i, a, b) for (int i = a; i < (b); ++i)
#define trav(a, x) for (auto& a : x)
#define sz(x) (int)(x).size()
typedef pair<int, int> pii;
typedef vector<int> vi;
#pragma GCC diagnostic ignored "-Wsign-compare"
// util functions
VI l;
VI x;
VI d;
VII use;
int n;
int c;
void cop(VI& a, vector<int32_t> b) {
a.resize(sz(b));
for (int i = 0; i < sz(a); i++) {
a[i] = b[i];
}
}
void genPref() {
x.resize(n);
x[0] = 0ll;
for (int i = 1; i < n; i++) {
x[i] = x[i - 1] + l[i - 1];
}
}
int mnsum, mxsum, mndif, mxdif;
void upd(int i) {
int x = ::x[i], d = ::d[i];
int sum = x + d, dif = x - d;
mnsum = min(mnsum, sum);
mxsum = max(mxsum, sum);
mndif = min(mndif, dif);
mxdif = max(mxdif, dif);
}
void initsumdif() {
mnsum = INF;
mxsum = -INF;
mndif = INF;
mxdif = -INF;
}
int K;
bool optimize(int i, int j) {
// do we need to optimize pair i and j?
return x[j] + d[j] - (x[i] - d[i]) > K;
}
bool works(int k) {
K = k;
VI vals = {INF, -INF, INF, -INF};
initsumdif();
int p1 = 0, p2 = 0;
for (int i = 0; i < n; i++) {
while (p1 < p2 and optimize(use[p1].y, i)) {
upd(use[p1++].y);
}
int sum = x[i] + d[i], dif = x[i] - d[i];
vals[0] = min(vals[0], (k - c) + dif + mndif);
vals[1] = max(vals[1], sum + mxsum - (k - c));
vals[2] = min(vals[2], (k - c) + dif - mxsum);
vals[3] = max(vals[3], sum - mndif - (k - c));
while (p2 > p1 and use[p2 - 1].x > x[i] - d[i]) p2--;
use[p2++] = MP(x[i] - d[i], i);
}
// now we need to find two points in the rectangle defined by vals
int maxsum, minsum, maxdif, mindif;
maxsum = vals[0];
minsum = vals[1];
maxdif = vals[2];
mindif = vals[3];
int curdif = 0;
int cursum = n;
if (maxsum < minsum or maxdif < mindif) return 0;
for (int i = 0; i < n; i++) {
while (curdif < n and x[curdif] - x[i] < mindif) curdif++;
while (cursum > 0 and x[cursum - 1] + x[i] >= minsum) cursum--;
int cur = max({cursum, curdif, i + 1});
if (cur < n and x[cur] + x[i] <= maxsum and x[cur] - x[i] <= maxdif)
return true;
}
return false;
}
ll find_shortcut(int32_t _n, vector<int32_t> _l, vector<int32_t> _d,
int32_t _c) {
n = _n;
c = _c;
cop(l, _l);
cop(d, _d);
genPref();
use.assign(n, MP(0, 0));
int xx = -1;
int z = INF;
for (int b = z; b >= 1; b /= 2) {
while (!works(xx + b)) xx += b;
}
return xx + 1;
}
