Problem:
Score: 400 points

Problem Statement

The Patisserie AtCoder sells cakes with number-shaped candles.
There are X, Y and Z kinds of cakes with 1-shaped, 2-shaped and 3-shaped candles, respectively.
Each cake has an integer value called deliciousness, as follows:

The deliciousness of the cakes with 1-shaped candles are A_1, A_2, ..., A_X.

The deliciousness of the cakes with 2-shaped candles are B_1, B_2, ..., B_Y.

The deliciousness of the cakes with 3-shaped candles are C_1, C_2, ..., C_Z.

Takahashi decides to buy three cakes, one for each of the three shapes of the candles, to celebrate ABC 123.

There are X \times Y \times Z such ways to choose three cakes.

We will arrange these X \times Y \times Z ways in descending order of the sum of the deliciousness of the cakes.

Print the sums of the deliciousness of the cakes for the first, second, ..., K-th ways in this list.

Constraints

1 \leq X \leq 1 \ 000

1 \leq Y \leq 1 \ 000

1 \leq Z \leq 1 \ 000

1 \leq K \leq \min(3 \ 000, X \times Y \times Z)

1 \leq A_i \leq 10 \ 000 \ 000 \ 000

1 \leq B_i \leq 10 \ 000 \ 000 \ 000

1 \leq C_i \leq 10 \ 000 \ 000 \ 000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

X Y Z K
A_1 \ A_2 \ A_3 \ ... \ A_X
B_1 \ B_2 \ B_3 \ ... \ B_Y
C_1 \ C_2 \ C_3 \ ... \ C_Z

Output

Print K lines. The i-th line should contain the i-th value stated in the problem statement.

Sample Input 1

2 2 2 8
4 6
1 5
3 8

Sample Output 1

19
17
15
14
13
12
10
8

There are 2 \times 2 \times 2 = 8 ways to choose three cakes, as shown below in descending order of the sum of the deliciousness of the cakes:

(A_2, B_2, C_2): 6 + 5 + 8 = 19

(A_1, B_2, C_2): 4 + 5 + 8 = 17

(A_2, B_1, C_2): 6 + 1 + 8 = 15

(A_2, B_2, C_1): 6 + 5 + 3 = 14

(A_1, B_1, C_2): 4 + 1 + 8 = 13

(A_1, B_2, C_1): 4 + 5 + 3 = 12

(A_2, B_1, C_1): 6 + 1 + 3 = 10

(A_1, B_1, C_1): 4 + 1 + 3 = 8

Sample Input 2

3 3 3 5
1 10 100
2 20 200
1 10 100

Sample Output 2

400
310
310
301
301

There may be multiple combinations of cakes with the same sum of the deliciousness. For example, in this test case, the sum of A_1, B_3, C_3 and the sum of A_3, B_3, C_1 are both 301.
However, they are different ways of choosing cakes, so 301 occurs twice in the output.

Sample Input 3

10 10 10 20
7467038376 5724769290 292794712 2843504496 3381970101 8402252870 249131806 6310293640 6690322794 6082257488
1873977926 2576529623 1144842195 1379118507 6003234687 4925540914 3902539811 3326692703 484657758 2877436338
4975681328 8974383988 2882263257 7690203955 514305523 6679823484 4263279310 585966808 3752282379 620585736

Sample Output 3

23379871545
22444657051
22302177772
22095691512
21667941469
21366963278
21287912315
21279176669
21160477018
21085311041
21059876163
21017997739
20703329561
20702387965
20590247696
20383761436
20343962175
20254073196
20210218542
20150096547

Note that the input or output may not fit into a 32-bit integer type.

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;

namespace abc123
{
    class ProgramD
    {
        public int K;
        public long[] A;
        public long[] B;
        public long[] C;

        static void Main(string[] args)
        {
            var program = new ProgramD();
            program.InitializeByConsole();
            // program.InitializeSample3();

            var outputs = program.GetOutput();

            foreach (var output in outputs)
            {
                Console.WriteLine(output);
            }
        }

        public void InitializeByConsole()
        {
            var nums = Console.ReadLine().Split(' ').Select(s => int.Parse(s)).ToArray();

            this.K = nums[3];
            this.A = Console.ReadLine().Split(' ').Select(s => long.Parse(s)).ToArray();
            this.B = Console.ReadLine().Split(' ').Select(s => long.Parse(s)).ToArray();
            this.C = Console.ReadLine().Split(' ').Select(s => long.Parse(s)).ToArray();
        }

        private static int GenIndexHash(int indexA, int indexB, int indexC)
        {
            return (indexA << 20) + (indexB << 10) + indexC;
        }

        private IEnumerable<long> GetOutput()
        {
            Array.Sort(this.A);
            Array.Sort(this.B);
            Array.Sort(this.C);

            Array.Reverse(this.A);
            Array.Reverse(this.B);
            Array.Reverse(this.C);

            var numa = Math.Min(this.K, this.A.Length);
            var numb = Math.Min(this.K, this.B.Length);
            var numc = Math.Min(this.K, this.C.Length);

            var abcIndexSet = new HashSet<int>();
            var abcQueue = new PriorityQueue<AbcSet>(new AbcTotalComparer());

            abcQueue.Enqueue(new AbcSet(0, 0, 0, this.A[0] + this.B[0] + this.C[0]));
            abcIndexSet.Add(GenIndexHash(0, 0, 0));
            var answers = new List<long>();

            while (true)
            {
                if (abcQueue.Count == 0)
                {
                    break;
                }

                var abc = abcQueue.Dequeue();
                if (answers.Count >= this.K)
                {
                    break;
                }

                answers.Add(abc.Result);

                if (abc.IndexA != numa - 1)
                {
                    Do(abc.IndexA + 1, abc.IndexB, abc.IndexC, abcQueue, abcIndexSet);
                }
                if (abc.IndexB != numb - 1)
                {
                    Do(abc.IndexA, abc.IndexB + 1, abc.IndexC, abcQueue, abcIndexSet);
                }
                if (abc.IndexC != numc - 1)
                {
                    Do(abc.IndexA, abc.IndexB, abc.IndexC + 1, abcQueue, abcIndexSet);
                }
            }

            return answers;
        }

        void Do(int ai, int bi, int ci, PriorityQueue<AbcSet> abcQueue, HashSet<int> abcIndexSet)
        {
            var hash = GenIndexHash(ai, bi, ci);
            if (abcIndexSet.Contains(hash))
            {
                return;
            }

            abcIndexSet.Add(hash);
            var total = this.A[ai] + this.B[bi] + this.C[ci];
            abcQueue.Enqueue(new AbcSet(ai, bi, ci, total));
        }

        public void InitializeSample1()
        {
            this.K = 8;
            this.A = new long[] {4, 6};
            this.B = new long[] {1, 5};
            this.C = new long[] {3, 8};
        }

        public void InitializeSample2()
        {
            this.K = 5;
            this.A = new long[] {1, 10, 100};
            this.B = new long[] {2, 20, 200};
            this.C = new long[] {1, 10, 100};
        }

        public void InitializeSample3()
        {
            this.K = 20;
            this.A = new[] {7467038376, 5724769290, 292794712, 2843504496, 3381970101, 8402252870, 249131806, 6310293640, 6690322794, 6082257488};
            this.B = new[] {1873977926, 2576529623, 1144842195, 1379118507, 6003234687, 4925540914, 3902539811, 3326692703, 484657758, 2877436338};
            this.C = new[] {4975681328, 8974383988, 2882263257, 7690203955, 514305523, 6679823484, 4263279310, 585966808, 3752282379, 620585736};
        }
    }

    public class AbcTotalComparer : IComparer<AbcSet>
    {
        public int Compare(AbcSet x, AbcSet y)
        {
            return y.Result.CompareTo(x.Result);
            var a = x.IndexA.CompareTo(y.IndexA);
            if (a != 0) return a;

            var b = x.IndexB.CompareTo(y.IndexB);
            if (b != 0) return b;

            return x.IndexC.CompareTo(y.IndexC);
        }
    }
    public struct AbcSet : IComparable<AbcSet>
    {
        public AbcSet(int indexA, int indexB, int indexC, long result)
        {
            IndexA = indexA;
            IndexB = indexB;
            IndexC = indexC;
            Result = result;
        }

        public int IndexA { get; }
        public int IndexB { get; }
        public int IndexC { get; }
        public long Result { get; }

        public int CompareTo(AbcSet other)
        {
            return this.Result.CompareTo(other.Result);
        }
    }

    #region PriorityQueue
    /// <summary>
    /// 指定した型のインスタンスを最も価値が低い順に取り出すことが可能な可変サイズのコレクションを表します．
    /// </summary>
    /// <typeparam name="T">優先度付きキュー内の要素の型を指定します．</typeparam>
    /// <remarks>内部的にはバイナリヒープによって実装されています．</remarks>
    public class PriorityQueue<T>
    {
        readonly List<T> heap = new List<T>();
        readonly Comparison<T> cmp;

        /// <summary>
        /// デフォルトの比較子を使用してインスタンスを初期化します．
        /// </summary>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public PriorityQueue() { cmp = Comparer<T>.Default.Compare; }

        /// <summary>
        /// デリゲートで表されるような比較関数を使用してインスタンスを初期化します．
        /// </summary>
        /// <param name="comparison"></param>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public PriorityQueue(Comparison<T> comparison) { cmp = comparison; }

        /// <summary>
        /// 指定された比較子を使用してインスタンスを初期化します．
        /// </summary>
        /// <param name="comparer"></param>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public PriorityQueue(IComparer<T> comparer) { cmp = comparer.Compare; }

        /// <summary>
        /// 優先度付きキューに要素を追加します．
        /// </summary>
        /// <param name="item">優先度付きキューに追加される要素</param>
        /// <remarks>最悪計算量 O(log N) で実行されます．</remarks>
        public void Enqueue(T item)
        {
            var pos = heap.Count;
            heap.Add(item);
            while (pos > 0)
            {
                var par = (pos - 1) / 2;
                if (cmp(heap[par], item) <= 0)
                    break;
                heap[pos] = heap[par];
                pos = par;
            }
            heap[pos] = item;

        }

        /// <summary>
        /// 優先度付きキューから最も価値が低い要素を削除し，返します．
        /// </summary>
        /// <returns>優先度付きキューから削除された要素．</returns>
        /// <remarks>最悪計算量 O(log N) で実行されます．</remarks>
        public T Dequeue()
        {
            var ret = heap[0];
            var pos = 0;
            var x = heap[heap.Count - 1];

            while (pos * 2 + 1 < heap.Count - 1)
            {
                var lch = pos * 2 + 1;
                var rch = pos * 2 + 2;
                if (rch < heap.Count - 1 && cmp(heap[rch], heap[lch]) < 0) lch = rch;
                if (cmp(heap[lch], x) >= 0)
                    break;
                heap[pos] = heap[lch];
                pos = lch;
            }
            heap[pos] = x;
            heap.RemoveAt(heap.Count - 1);
            return ret;

        }
        /// <summary>
        ///  優先度付きキューに含まれる最も価値が低い要素を削除せずに返します．
        /// </summary>
        /// <returns>優先度付きキューに含まれる最も価値が低い要素．</returns>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public T Peek() { return heap[0]; }

        /// <summary>
        /// 優先度付きキュー内の要素の数を取得します．
        /// </summary>
        /// <returns>優先度付キュー内にある要素の数</returns>
        /// <remarks>最悪計算量 O(1) で実行されます．</remarks>
        public int Count { get { return heap.Count; } }
        /// <summary>
        /// 優先度付きキュー内に要素が存在するかどうかを O(1) で判定します．
        /// </summary>
        /// <returns>優先度付キュー内にある要素が存在するならば true，そうでなければ　false．</returns>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public bool Any() { return heap.Count > 0; }

        /// <summary>
        /// 優先度付きキューに含まれる要素を昇順に並べて返します．
        /// </summary>
        /// <remarks>この操作は計算量 O(N log N)で実行されます．</remarks>
        public T[] Items
        {
            get
            {
                var ret = heap.ToArray();
                Array.Sort(ret, cmp);
                return ret;
            }
        }
    }
    #endregion
}