#include "shortcut.h"
#include <bits/stdc++.h>
typedef long long ll;
using namespace std;
const ll INF = 1e15;
ll p[1000000], half_plane[4];
pair<ll, int> ord1[1000000], ord2[1000000];
ll find_shortcut(int n, vector<int> l, vector<int> d, int c) {
for (int i = 1; i < n; i++) p[i] = p[i - 1] + l[i - 1];
// Order by d[i] + p[i] and d[i] - p[i]
for (int i = 0; i < n; i++)
ord1[i] = {d[i] + p[i], i}, ord2[i] = {d[i] - p[i], i};
sort(ord1, ord1 + n);
sort(ord2, ord2 + n);
// Binary search for the answer
ll ans = 0;
for (ll i = 1ll << 48; i; i >>= 1) {
ll mid = ans + i;
// Reset maximums and minimums
half_plane[0] = half_plane[1] = half_plane[2] = half_plane[3] = -INF;
ll mx1 = -INF, mx2 = -INF, smx1 = -INF, smx2 = -INF;
for (int i = 0, j = n - 1; i < n; i++) {
while (~j && ord1[i].first + ord2[j].first > mid) {
ll tmp1 = d[ord2[j].second] + p[ord2[j].second];
if (tmp1 > mx1) smx1 = mx1, mx1 = tmp1;
else if (tmp1 > smx1) smx1 = tmp1;
ll tmp2 = d[ord2[j].second] - p[ord2[j].second];
if (tmp2 > mx2) smx2 = mx2, mx2 = tmp2;
else if (tmp2 > smx2) smx2 = tmp2;
j--;
}
// Update the half planes
ll use1 = (d[ord1[i].second] + p[ord1[i].second] == mx1 ? smx1 : mx1);
ll use2 = (d[ord1[i].second] - p[ord1[i].second] == mx2 ? smx2 : mx2);
half_plane[0] = max(half_plane[0], c - mid + d[ord1[i].second] + p[ord1[i].second] + use1);
half_plane[1] = max(half_plane[1], c - mid + d[ord1[i].second] - p[ord1[i].second] + use1);
half_plane[2] = max(half_plane[2], c - mid + d[ord1[i].second] + p[ord1[i].second] + use2);
half_plane[3] = max(half_plane[3], c - mid + d[ord1[i].second] - p[ord1[i].second] + use2);
}
// Check that the half-plane intersection contains a good point
bool bad = true;
// Point (x, y) is below the half-plane intersection if y < max(-x + half_plane[0], x + half_plane[1])
// Point (x, y) is above the half-plane intersection if y > min(-x - half_plane[3], x - half_plane[2])
for (int i = 0, lower = n - 1, upper = 0; i < n; i++) {
while (~lower && p[lower] >= max(-p[i] + half_plane[0], p[i] + half_plane[1]))
lower--;
while (lower + 1 < n && p[lower + 1] < max(-p[i] + half_plane[0], p[i] + half_plane[1]))
lower++;
while (upper < n && p[upper] <= min(-p[i] - half_plane[3], p[i] - half_plane[2]))
upper++;
while (upper && p[upper - 1] > min(-p[i] - half_plane[3], p[i] - half_plane[2]))
upper--;
if (upper - lower > 1) {
bad = false;
break;
}
}
if (bad) ans = mid;
}
return ans + 1;
}
