Problem:
Score : 1200 points

Problem Statement

#nck {
width: 30px;
height: auto;
}

Imagine a game played on a line. Initially, the player is located at position 0 with N candies in his possession, and the exit is at position E. There are also N bears in the game. The i-th bear is located at x_i. The maximum moving speed of the player is 1 while the bears do not move at all.

When the player gives a candy to a bear, it will provide a coin after T units of time. More specifically, if the i-th bear is given a candy at time t, it will put a coin at its position at time t+T. The purpose of this game is to give candies to all the bears, pick up all the coins, and go to the exit. Note that the player can only give a candy to a bear if the player is at the exact same position of the bear. Also, each bear will only produce a coin once. If the player visits the position of a coin after or at the exact same time that the coin is put down, the player can pick up the coin. Coins do not disappear until collected by the player.

Shik is an expert of this game. He can give candies to bears and pick up coins instantly. You are given the configuration of the game. Please calculate the minimum time Shik needs to collect all the coins and go to the exit.

Constraints

1 \leq N \leq 100,000

1 \leq T, E \leq 10^9

0 < x_i < E

x_i < x_{i+1} for 1 \leq i < N

All input values are integers.

Partial Scores

In test cases worth 600 points, N \leq 2,000.

Input

The input is given from Standard Input in the following format:

N E T
x_1 x_2 ... x_N

Output

Print an integer denoting the answer.

Sample Input 1

3 9 1
1 3 8

Sample Output 1

12

The optimal strategy is to wait for the coin after treating each bear. The total time spent on waiting is 3 and moving is 9. So the answer is 3 + 9 = 12.

Sample Input 2

3 9 3
1 3 8

Sample Output 2

16

Sample Input 3

2 1000000000 1000000000
1 999999999

Sample Output 3

2999999996

Solution in C (GCC 5.4.1) (judge verdict: Accepted)

/*
cat <<EOF >mistaken-paste
*/

#pragma GCC diagnostic ignored "-Wincompatible-pointer-types"
#define _USE_MATH_DEFINES

#include <stdio.h>
#include <stdint.h>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>
#include <math.h>
#include <time.h>

#define BIG 2000000007
#define VERYBIG 2000000000000007LL

#define MOD 1000000007
#define FOD  998244353
typedef uint64_t ull;
typedef  int64_t sll;

#define N_MAX 1000000
#define M_MAX 200000

#ifdef __cplusplus
#include <queue>
#include <stack>
#include <tuple>
#include <set>
#include <map>
#include <string>
#include <algorithm>
#include <functional>
#include <array>

using std::queue;
using std::priority_queue;
using std::stack;
using std::tuple;
using std::set;
using std::map;
using std::vector;
using std::greater;
using std::pair;
using std::string;

#endif

typedef struct {
	int32_t a;
	int32_t b;
} hw;

typedef struct {
	sll a;
	sll b;
} hwll;

typedef struct {
	sll a;
	sll b;
	sll c;
} hwllc;

typedef struct {
	hwll a;
	hwll b;
} linell;

ull n, m;
ull h, w;
ull k;
ull q;
sll va, vb, vc, vd, ve, vf;
ull ua, ub, uc, ud, ue, uf;
long double vra, vrb, vrc;
double vda, vdb, vdc;
char ch, dh;

ull umin (ull x, ull y) {
	return (x < y) ? x : y;
}

ull umax (ull x, ull y) {
	return (x > y) ? x : y;
}

sll smin (sll x, sll y) {
	return (x < y) ? x : y;
}

sll smax (sll x, sll y) {
	return (x > y) ? x : y;
}

ull gcd (ull x, ull y) {
	if (x < y) {
		return gcd(y, x);
	} else if (y == 0) {
		return x;
	} else {
		return gcd(y, x % y);
	}
}

ull bitpow (ull a, ull x, ull modulo) {
	ull result = 1;
	while (x) {
		if (x & 1) {
			result *= a;
			result %= modulo;
		}
		x /= 2;
		a = (a * a) % modulo;
	}
	return result;
}

ull divide (ull a, ull b, ull modulo) {
	return (a * bitpow(b, modulo - 2, modulo)) % modulo;
}

ull udiff (ull a, ull b) {
	if (a >= b) {
		return a - b;
	} else {
		return b - a;
	}
}

sll sdiff (sll a, sll b) {
	if (a >= b) {
		return a - b;
	} else {
		return b - a;
	}
}

int bitcount (ull n) {
	int result = 0;
	while (n) {
		if (n & 1) result++;
		n /= 2;
	}
	return result;
}

// double distance (sll x1, sll y1, sll x2, sll y2) {
// 	double xdist2, ydist2, origindist, dist;

// 	xdist2 = (x1 - x2) * (x1 - x2);
// 	ydist2 = (y1 - y2) * (y1 - y2);
// 	return sqrt(xdist2 + ydist2);
// }

int32_t pullcomp (const void *left, const void *right) {
	ull l = *(ull*)left;
	ull r = *(ull*)right;
	if (l < r) {
		return -1;
	}
	if (l > r) {
		return +1;
	}
	return 0;
}

int32_t psllcomp (const void *left, const void *right) {
	sll l = *(sll*)left;
	sll r = *(sll*)right;
	if (l < r) {
		return -1;
	}
	if (l > r) {
		return +1;
	}
	return 0;
}

int32_t pcharcomp (const void *left, const void *right) {
	char l = *(char*)left;
	char r = *(char*)right;
	if (l < r) {
		return -1;
	}
	if (l > r) {
		return +1;
	}
	return 0;
}

int32_t pdoublecomp (const void *left, const void *right) {
	double l = *(double*)left;
	double r = *(double*)right;
	if (l < r) {
		return -1;
	}
	if (l > r) {
		return +1;
	}
	return 0;
}

int32_t pstrcomp (const void *left, const void *right) {
	char* l = *(char**)left;
	char* r = *(char**)right;

	return strcmp(l, r);
}

int32_t phwllABcomp (const void *left, const void *right) {
	hwll l = *(hwll*)left;
	hwll r = *(hwll*)right;
	if (l.a < r.a) {
		return -1;
	}
	if (l.a > r.a) {
		return +1;
	}
	if (l.b < r.b) {
		return -1;
	}
	if (l.b > r.b) {
		return +1;
	}
	return 0;
}

int32_t phwllREVcomp (const void *left, const void *right) {
	hwll l = *(hwll*)left;
	hwll r = *(hwll*)right;
	if (l.b < r.b) {
		return -1;
	}
	if (l.b > r.b) {
		return +1;
	}
	if (l.a < r.a) {
		return -1;
	}
	if (l.a > r.a) {
		return +1;
	}
	return 0;
}

int32_t ptriplecomp (const void *left, const void *right) {
	hwllc l = *(hwllc*)left;
	hwllc r = *(hwllc*)right;

	if (l.a < r.a) {
		return -1;
	}
	if (l.a > r.a) {
		return +1;
	}
	if (l.b < r.b) {
		return -1;
	}
	if (l.b > r.b) {
		return +1;
	}
	if (l.c < r.c) {
		return -1;
	}
	if (l.c > r.c) {
		return +1;
	}
	return 0;
}

bool isinrange (sll left, sll x, sll right) {
	return (left <= x && x <= right);
}

bool isinrange_soft (sll left, sll x, sll right) {
	return (left <= x && x <= right) || (left >= x && x >= right);
}

sll a[N_MAX];
// sll a[3001][3001];
sll b[N_MAX];
// sll b[3001][3001];
sll c[N_MAX];
// sll d[N_MAX];
// sll e[N_MAX];
// char s[N_MAX + 1];
char s[3010][3010];
// char t[N_MAX + 1];
char t[3010][3010];
// hwll xy[N_MAX];
// sll table[1000][1000];

ull dp1[N_MAX];
ull dp2[N_MAX];

void smaller (ull *x, ull y) {
	if (*x > y) *x = y;
}

ull solve () {
	sll i, j, ki, li;
	ull result = 0;
	// sll result = 0;
	double dresult = 0;
	// ull maybe = 0;
	sll maybe = 0;
	// ull sum = 0;
	sll sum = 0;
	sll item;
	ull *dpcell;

	for (i = 0; i <= n; i++) {
		dp1[i] = dp2[i] = VERYBIG;
	}
	dp1[0] = 0;

	for (i = 0; i < n; i++) {
		ull ok = i, ng = n + 1;
		while (ok + 1 < ng) {
			ull med = (ok + ng) / 2;
			if (2 * (a[med] - a[i + 1]) < k) {
				ok = med;
			} else {
				ng = med;
			}
		}

		smaller(&dp2[i + 1], dp2[i] + 3 * (a[i + 1] - a[i]));
		ull x = umin(dp1[i], dp2[i]);
		smaller(&dp1[ok], x + k + (a[ok] - a[i]));
		smaller(&dp2[ng], x + 3 * (a[ng] - a[i + 1]) + (a[i + 1] - a[i]));
	}

	result = umin(dp1[n], dp2[n]) + (m - a[n]);

	printf("%lld\n", result);
	// printf("%.15lf\n", dresult);
	// puts(s);

	return 0;

	success:
	// puts("YES");
	puts("Yes");
	// printf("%llu\n", result);
	// puts("0");
	// puts("Yay!");
	return 0;

	fail:
	// puts("NO");
	// puts("No");
	puts("0");
	// puts("-1");
	// puts("-1 -1 -1");
	// puts(":(");
	return 1;
}

int32_t main (void) {
	int32_t i, j;
	int32_t x, y;

	// scanf("%llu%llu", &h, &w);
	scanf("%llu%llu", &n, &m);
	scanf("%llu", &k, &n, &m);
	// scanf("%llu%llu", &h, &w);
	// scanf("%lld", &va, &vb, &vc, &vd);
	// scanf("%llu%llu", &ua, &ub, &uc, &ud);
	// scanf("%llu", &q);
	// scanf("%s", s);
	// scanf("%s", t);
	for (i = 0; i < n; i++) {
		// scanf("%lld%lld", &xy[i].a, &xy[i].b);
		scanf("%lld", &a[i + 1]);
		// scanf("%lld", &b[i]);
		// scanf("%lld", &c[i]);
		// scanf("%lld", &d[i]);
		// a[i]--;
		// b[i]--;
		// c[i]--;
	}
	// scanf("%llu", &m, &k);
	// scanf("%llu", &q);
	// for (i = 0; i < n; i++) {
	// 	scanf("%lld", &b[i]);
	// 	b[i]--;
	// }

	// for (i = 0; i < h; i++) {
	// 	for (j = 0; j < w; j++) {
	// 		scanf("%llu", &table[i][j]);
	// 	}
	// }
	// for (i = 0; i < h; i++) {
	// 	scanf("%s", &s[i]);
	// }

	solve();

	return 0;
}