Problem:
Score : 500 points

Problem Statement

"Teishi-zushi", a Japanese restaurant, is a plain restaurant with only one round counter. The outer circumference of the counter is C meters. Customers cannot go inside the counter.

Nakahashi entered Teishi-zushi, and he was guided to the counter. Now, there are N pieces of sushi (vinegared rice with seafood and so on) on the counter. The distance measured clockwise from the point where Nakahashi is standing to the point where the i-th sushi is placed, is x_i meters. Also, the i-th sushi has a nutritive value of v_i kilocalories.

Nakahashi can freely walk around the circumference of the counter. When he reach a point where a sushi is placed, he can eat that sushi and take in its nutrition (naturally, the sushi disappears). However, while walking, he consumes 1 kilocalories per meter.

Whenever he is satisfied, he can leave the restaurant from any place (he does not have to return to the initial place). On balance, at most how much nutrition can he take in before he leaves? That is, what is the maximum possible value of the total nutrition taken in minus the total energy consumed? Assume that there are no other customers, and no new sushi will be added to the counter. Also, since Nakahashi has plenty of nutrition in his body, assume that no matter how much he walks and consumes energy, he never dies from hunger.

Constraints

1 ≤ N ≤ 10^5

2 ≤ C ≤ 10^{14}

1 ≤ x_1 < x_2 < ... < x_N < C

1 ≤ v_i ≤ 10^9

All values in input are integers.

Subscores

300 points will be awarded for passing the test set satisfying N ≤ 100.

Input

Input is given from Standard Input in the following format:

N C
x_1 v_1
x_2 v_2
:
x_N v_N

Output

If Nakahashi can take in at most c kilocalories on balance before he leaves the restaurant, print c.

Sample Input 1

3 20
2 80
9 120
16 1

Sample Output 1

191

There are three sushi on the counter with a circumference of 20 meters. If he walks two meters clockwise from the initial place, he can eat a sushi of 80 kilocalories. If he walks seven more meters clockwise, he can eat a sushi of 120 kilocalories. If he leaves now, the total nutrition taken in is 200 kilocalories, and the total energy consumed is 9 kilocalories, thus he can take in 191 kilocalories on balance, which is the largest possible value.

Sample Input 2

3 20
2 80
9 1
16 120

Sample Output 2

192

The second and third sushi have been swapped. Again, if he walks two meters clockwise from the initial place, he can eat a sushi of 80 kilocalories. If he walks six more meters counterclockwise this time, he can eat a sushi of 120 kilocalories. If he leaves now, the total nutrition taken in is 200 kilocalories, and the total energy consumed is 8 kilocalories, thus he can take in 192 kilocalories on balance, which is the largest possible value.

Sample Input 3

1 100000000000000
50000000000000 1

Sample Output 3

0

Even though the only sushi is so far that it does not fit into a 32-bit integer, its nutritive value is low, thus he should immediately leave without doing anything.

Sample Input 4

15 10000000000
400000000 1000000000
800000000 1000000000
1900000000 1000000000
2400000000 1000000000
2900000000 1000000000
3300000000 1000000000
3700000000 1000000000
3800000000 1000000000
4000000000 1000000000
4100000000 1000000000
5200000000 1000000000
6600000000 1000000000
8000000000 1000000000
9300000000 1000000000
9700000000 1000000000

Sample Output 4

6500000000

All these sample inputs above are included in the test set for the partial score.

Solution in C# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.IO;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;
using System.Security.Principal;

namespace AtCoder.Contest.E
{
	static class Program
	{
		public static void Solve(Scanner cin)
		{
			var (n, c) = cin.ReadValue<int, long>();
			var (_x, _v) = cin.ReadValueArray<long, long>(n);
			var x = new List<long>();
			var v = new List<long>();
			x.Add(0); v.Add(0);
			for (int i = 0; i < n; i++)
			{
				x.Add(_x[i]); v.Add(_v[i]);
			}
			x.Add(c); v.Add(0);

			var sumleft = new long[n + 2];
			var sumright = new long[n + 2];
			for (int i = 0; i < n + 2; i++)
			{
				if (i == 0)
				{
					sumleft[i] = v[i];
					sumright[i] = v[n + 1 - i];
				}
				else
				{
					sumleft[i] = v[i] + sumleft[i - 1];
					sumright[i] = v[n + 1 - i] + sumright[i - 1];
				}
			}
			var l1 = new long[n + 2];
			var l2 = new long[n + 2];
			var r1 = new long[n + 2];
			var r2 = new long[n + 2];

			for (int i = 0; i < n+2; i++)
			{
				l1[i] = sumleft[i] - x[i];
				l2[i] = sumleft[i] - x[i] * 2;
				r1[i] = sumright[i] - (c - x[n + 1 - i]);
				r2[i] = sumright[i] - (c - x[n + 1 - i]) * 2;
			}

			var l1s = new SegmentTree(l1);
			var l2s = new SegmentTree(l2);
			var r1s = new SegmentTree(r1);
			var r2s = new SegmentTree(r2);

			long ans = 0;

			for (int i = 1; i < n + 1; i++)
			{
				ans = Math.Max(ans, l1[i] + r2s.GetMax(0, n - i));
				ans = Math.Max(ans, r1[i] + l2s.GetMax(0, n - i));

				ans = Math.Max(ans, l2[i] + r1s.GetMax(0, n - i));
				ans = Math.Max(ans, r2[i] + l1s.GetMax(0, n - i));
			}

			Console.WriteLine(ans);
		}

		public static void Main(string[] args)
		{
			var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
			Console.SetOut(sw);
			var cin = new Scanner();
			Solve(cin);
			Console.Out.Flush();
		}

		public static void YESNO(bool condition)
		{
			if (condition) Console.WriteLine("YES");
			else Console.WriteLine("NO");
		}

		public static void YesNo(bool condition)
		{
			if (condition) Console.WriteLine("Yes");
			else Console.WriteLine("No");
		}

		public static void yesno(bool condition)
		{
			if (condition) Console.WriteLine("yes");
			else Console.WriteLine("no");
		}
	}

	class SegmentTree
	{
		// 元の配列の大きさ
		public int N { get; set; }
		// セグメント木のデータ(2^n-1)
		public long[] Node { get; set; }

		// 元データの配列を引数で渡す
		public SegmentTree(long[] a)
		{
			// 要素数を2のべき乗とする
			N = 1;
			while (N < a.Length) N *= 2;

			// セグメント木の管理する要素数を 2^n-1 で初期化する
			this.Node = new long[2 * N - 1];
			for (int i = 0; i < this.Node.Length; i++) this.Node[i] = -1;

			// 各ノードの値を初期化する
			// 葉は元の配列のデータ
			for (int i = 0; i < a.Length; i++) this.Node[N + i - 1] = a[i];
			// 子ノードを比較して親ノードの値を決めていく
			for (int i = N - 2; i >= 0; i--)
			{
				this.Node[i] = this.Compare(this.Node[2 * i + 1], this.Node[2 * i + 2]);
			}
		}

		// x番目のデータをvに更新する
		public void Update(int x, int v)
		{
			// 元の配列のデータが格納される葉のデータを更新する
			x += (this.N - 1);
			this.Node[x] = v;

			// 更新したノードの親をさかのぼってルートまで更新する
			while (x > 0)
			{
				// 親ノードのインデックス
				x = (x - 1) / 2;
				this.Node[x] = this.Compare(this.Node[2 * x + 1], this.Node[2 * x + 2]);
			}
		}
		// [a, b) の区間最小値を求める
		public long GetMax(int a, int b)
		{
			return this.GetMax(a, b, 0, 0, this.N);
		}
		// [a, b) の区間最小値を求める
		// 現在のノードはk番目、[l, r)
		private long GetMax(int a, int b, int k, long l, long r)
		{
			// 1. 現在のノードの対応区間が、求める区間に含まれない場合
			//    対象外の区間のデータなので、最小値に関係のない値(INF)を返す
			if (r <= a || b <= l) return -1;

			// 2. 現在のノードの対応区間が、求める区間に完全に含まれる場合
			//    現在のノードの値が最小値の候補になるのでそのまま返す
			if (a <= l && r <= b) return this.Node[k];

			// 3. 上記以外の場合、現在のノードの対応区間が、求める区間を一部を含む場合
			//    子ノードを評価し、比較した結果を返す
			var vl = this.GetMax(a, b, 2 * k + 1, l, (l + r) / 2);
			var vr = this.GetMax(a, b, 2 * k + 2, (l + r) / 2, r);
			return this.Compare(vl, vr);
		}

		// 2要素を比較して小さい値を返す
		private long Compare(long a, long b) { return Math.Max(a, b); }
	}

	class Scanner
	{
		string[] s;
		int i;

		char[] separator = new char[] { ' ' };

		public Scanner()
		{
			s = new string[0];
			i = 0;
		}

		public string Read() => ReadString();

		public string ReadString()
		{
			if (i < s.Length) return s[i++];
			string st = Console.ReadLine();
			while (st == "") st = Console.ReadLine();
			s = st.Split(separator, StringSplitOptions.RemoveEmptyEntries);
			if (s.Length == 0) return ReadString();
			i = 0;
			return s[i++];
		}

		public string[] ReadStringArray(int N)
		{
			string[] Array = new string[N];
			for (int i = 0; i < N; i++)
			{
				Array[i] = ReadString();
			}
			return Array;
		}

		public int ReadInt()
		{
			return int.Parse(ReadString());
		}

		public int[] ReadIntArray(int N, int add = 0)
		{
			int[] Array = new int[N];
			for (int i = 0; i < N; i++)
			{
				Array[i] = ReadInt() + add;
			}
			return Array;
		}

		public long ReadLong()
		{
			return long.Parse(ReadString());
		}

		public long[] ReadLongArray(int N, long add = 0)
		{
			long[] Array = new long[N];
			for (int i = 0; i < N; i++)
			{
				Array[i] = ReadLong() + add;
			}
			return Array;
		}

		public double ReadDouble()
		{
			return double.Parse(ReadString());
		}

		public double[] ReadDoubleArray(int N, double add = 0)
		{
			double[] Array = new double[N];
			for (int i = 0; i < N; i++)
			{
				Array[i] = ReadDouble() + add;
			}
			return Array;
		}

		public T1 ReadValue<T1>() => (T1)Convert.ChangeType(ReadString(), typeof(T1));

		public (T1, T2) ReadValue<T1, T2>()
		{
			var inputs = ReadStringArray(2);
			var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
			var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
			return (v1, v2);
		}

		public (T1, T2, T3) ReadValue<T1, T2, T3>()
		{
			var inputs = ReadStringArray(3);
			var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
			var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
			var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
			return (v1, v2, v3);
		}

		public (T1, T2, T3, T4) ReadValue<T1, T2, T3, T4>()
		{
			var inputs = ReadStringArray(4);
			var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
			var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
			var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
			var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4));
			return (v1, v2, v3, v4);
		}

		public (T1, T2, T3, T4, T5) ReadValue<T1, T2, T3, T4, T5>()
		{
			var inputs = ReadStringArray(5);
			var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
			var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
			var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
			var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4));
			var v5 = (T5)Convert.ChangeType(inputs[4], typeof(T5));
			return (v1, v2, v3, v4, v5);
		}

		public (T1, T2, T3, T4, T5, T6) ReadValue<T1, T2, T3, T4, T5, T6>()
		{
			var inputs = ReadStringArray(6);
			var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
			var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
			var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
			var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4));
			var v5 = (T5)Convert.ChangeType(inputs[4], typeof(T5));
			var v6 = (T6)Convert.ChangeType(inputs[5], typeof(T6));
			return (v1, v2, v3, v4, v5, v6);
		}

		public (T1, T2, T3, T4, T5, T6, T7) ReadValue<T1, T2, T3, T4, T5, T6, T7>()
		{
			var inputs = ReadStringArray(7);
			var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
			var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
			var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
			var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4));
			var v5 = (T5)Convert.ChangeType(inputs[4], typeof(T5));
			var v6 = (T6)Convert.ChangeType(inputs[5], typeof(T6));
			var v7 = (T7)Convert.ChangeType(inputs[6], typeof(T7));
			return (v1, v2, v3, v4, v5, v6, v7);
		}

		public T1[] ReadValueArray<T1>(int N)
		{
			var v1 = new T1[N];
			for (int i = 0; i < N; i++)
			{
				v1[i] = ReadValue<T1>();
			}
			return v1;
		}

		public (T1[], T2[]) ReadValueArray<T1, T2>(int N)
		{
			var (v1, v2) = (new T1[N], new T2[N]);
			for (int i = 0; i < N; i++)
			{
				var (t1, t2) = ReadValue<T1, T2>();
				v1[i] = t1;
				v2[i] = t2;
			}
			return (v1, v2);
		}

		public (T1[], T2[], T3[]) ReadValueArray<T1, T2, T3>(int N)
		{
			var (v1, v2, v3) = (new T1[N], new T2[N], new T3[N]);
			for (int i = 0; i < N; i++)
			{
				var (t1, t2, t3) = ReadValue<T1, T2, T3>();
				v1[i] = t1;
				v2[i] = t2;
				v3[i] = t3;
			}
			return (v1, v2, v3);
		}

		public (T1[], T2[], T3[], T4[]) ReadValueArray<T1, T2, T3, T4>(int N)
		{
			var (v1, v2, v3, v4) = (new T1[N], new T2[N], new T3[N], new T4[N]);
			for (int i = 0; i < N; i++)
			{
				var (t1, t2, t3, t4) = ReadValue<T1, T2, T3, T4>();
				v1[i] = t1;
				v2[i] = t2;
				v3[i] = t3;
				v4[i] = t4;
			}
			return (v1, v2, v3, v4);
		}

		public (T1[], T2[], T3[], T4[], T5[]) ReadValueArray<T1, T2, T3, T4, T5>(int N)
		{
			var (v1, v2, v3, v4, v5) = (new T1[N], new T2[N], new T3[N], new T4[N], new T5[N]);
			for (int i = 0; i < N; i++)
			{
				var (t1, t2, t3, t4, t5) = ReadValue<T1, T2, T3, T4, T5>();
				v1[i] = t1;
				v2[i] = t2;
				v3[i] = t3;
				v4[i] = t4;
				v5[i] = t5;
			}
			return (v1, v2, v3, v4, v5);
		}

		public (T1[], T2[], T3[], T4[], T5[], T6[]) ReadValueArray<T1, T2, T3, T4, T5, T6>(int N)
		{
			var (v1, v2, v3, v4, v5, v6) = (new T1[N], new T2[N], new T3[N], new T4[N], new T5[N], new T6[N]);
			for (int i = 0; i < N; i++)
			{
				var (t1, t2, t3, t4, t5, t6) = ReadValue<T1, T2, T3, T4, T5, T6>();
				v1[i] = t1;
				v2[i] = t2;
				v3[i] = t3;
				v4[i] = t4;
				v5[i] = t5;
				v6[i] = t6;
			}
			return (v1, v2, v3, v4, v5, v6);
		}

		public (T1[], T2[], T3[], T4[], T5[], T6[], T7[]) ReadValueArray<T1, T2, T3, T4, T5, T6, T7>(int N)
		{
			var (v1, v2, v3, v4, v5, v6, v7) = (new T1[N], new T2[N], new T3[N], new T4[N], new T5[N], new T6[N], new T7[N]);
			for (int i = 0; i < N; i++)
			{
				var (t1, t2, t3, t4, t5, t6, t7) = ReadValue<T1, T2, T3, T4, T5, T6, T7>();
				v1[i] = t1;
				v2[i] = t2;
				v3[i] = t3;
				v4[i] = t4;
				v5[i] = t5;
				v6[i] = t6;
				v7[i] = t7;
			}
			return (v1, v2, v3, v4, v5, v6, v7);
		}
	}
}