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)

#if DEBUG
using Microsoft.VisualStudio.TestTools.UnitTesting;
#endif
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace Contest
{
    class Program
    {
        public static IO.StreamScanner scanner = new IO.StreamScanner(Console.OpenStandardInput());

        static void Main(string[] args)
        {
            var x = scanner.Integer();
            var y = scanner.Integer();
            var z = scanner.Integer();
            var k = scanner.Integer();
            var a = scanner.ScanLine().Split().Select(v => long.Parse(v)).ToList();
            var b = scanner.ScanLine().Split().Select(v => long.Parse(v)).ToList();
            var c = scanner.ScanLine().Split().Select(v => long.Parse(v)).ToList();

            var ans = GetAns(k, x, y, z, a, b, c);
            foreach (var r in ans) IO.Printer.Out.WriteLine(r);

            IO.Printer.Out.Flush();
        }

        public static List<long> GetAns(int k, int x, int y, int z,
            List<long> a, List<long> b, List<long> c)
        {
            var ret = new List<long>();
            a = a.OrderByDescending(v => v).ToList();
            b = b.OrderByDescending(v => v).ToList();
            c = c.OrderByDescending(v => v).ToList();

            var que = new PriorityQueue<Node>();
            que.Push(new Node(a, b, c) { A_Idx = 0, B_Idx = 0, C_Idx = 0 });

            var used = new HashSet<Tuple<int, int, int>>();
            used.Add(new Tuple<int, int, int>(0, 0, 0));

            while (ret.Count() < k)
            {
                var cur = que.Pop();
                ret.Add(cur.Sum());

                if (cur.A_Idx < a.Count() - 1 && !used.Contains(new Tuple<int, int, int>(cur.A_Idx + 1, cur.B_Idx, cur.C_Idx)))
                {
                    used.Add(new Tuple<int, int, int>(cur.A_Idx + 1, cur.B_Idx, cur.C_Idx));
                    que.Push(new Node(a, b, c) { A_Idx = cur.A_Idx + 1, B_Idx = cur.B_Idx, C_Idx = cur.C_Idx });
                }
                if (cur.B_Idx < b.Count() - 1 && !used.Contains(new Tuple<int, int, int>(cur.A_Idx, cur.B_Idx + 1, cur.C_Idx)))
                {
                    used.Add(new Tuple<int, int, int>(cur.A_Idx, cur.B_Idx + 1, cur.C_Idx));
                    que.Push(new Node(a, b, c) { A_Idx = cur.A_Idx, B_Idx = cur.B_Idx + 1, C_Idx = cur.C_Idx });
                }
                if (cur.C_Idx < c.Count() - 1 && !used.Contains(new Tuple<int, int, int>(cur.A_Idx, cur.B_Idx, cur.C_Idx + 1)))
                {
                    used.Add(new Tuple<int, int, int>(cur.A_Idx, cur.B_Idx, cur.C_Idx + 1));
                    que.Push(new Node(a, b, c) { A_Idx = cur.A_Idx, B_Idx = cur.B_Idx, C_Idx = cur.C_Idx + 1 });
                }
            }

            return ret;
        }
    }

    public class Node : IComparable
    {
        public int A_Idx;
        public int B_Idx;
        public int C_Idx;

        List<long> _a;
        List<long> _b;
        List<long> _c;

        public Node(List<long> a, List<long> b, List<long> c)
        {
            _a = a;
            _b = b;
            _c = c;
        }

        public int CompareTo(object obj)
        {
            var other = obj as Node;
            return other.Sum().CompareTo(Sum());
        }

        public long Sum()
        {
            return _a[A_Idx] + _b[B_Idx] + _c[C_Idx];
        }
    }

    public class PriorityQueue<T> where T : IComparable
    {
        private IComparer<T> _comparer = null;
        private int _type = 0;

        private T[] _heap;
        private int _sz = 0;

        private int _count = 0;

        /// <summary>
        /// Priority Queue with custom comparer
        /// </summary>
        public PriorityQueue(IComparer<T> comparer)
        {
            _heap = new T[128];
            _comparer = comparer;
        }

        /// <summary>
        /// Priority queue
        /// </summary>
        /// <param name="type">0: asc, 1:desc</param>
        public PriorityQueue(int type = 0)
        {
            _heap = new T[128];
            _type = type;
        }

        private int Compare(T x, T y)
        {
            if (_comparer != null) return _comparer.Compare(x, y);
            return _type == 0 ? x.CompareTo(y) : y.CompareTo(x);
        }

        public void Push(T x)
        {
            _count++;
            if (_count > _heap.Length)
            {
                var newheap = new T[_heap.Length * 2];
                for (int n = 0; n < _heap.Length; n++) newheap[n] = _heap[n];
                _heap = newheap;
            }

            //node number
            var i = _sz++;

            while (i > 0)
            {
                //parent node number
                var p = (i - 1) / 2;

                if (Compare(_heap[p], x) <= 0) break;

                _heap[i] = _heap[p];
                i = p;
            }

            _heap[i] = x;
        }

        public T Pop()
        {
            _count--;

            T ret = _heap[0];
            T x = _heap[--_sz];

            int i = 0;
            while (i * 2 + 1 < _sz)
            {
                //children
                int a = i * 2 + 1;
                int b = i * 2 + 2;

                if (b < _sz && Compare(_heap[b], _heap[a]) < 0) a = b;

                if (Compare(_heap[a], x) >= 0) break;

                _heap[i] = _heap[a];
                i = a;
            }

            _heap[i] = x;

            return ret;
        }

        public int Count()
        {
            return _count;
        }

        public T Peek()
        {
            return _heap[0];
        }

        public bool Contains(T x)
        {
            for (int i = 0; i < _sz; i++) if (x.Equals(_heap[i])) return true;
            return false;
        }

        public void Clear()
        {
            while (this.Count() > 0) this.Pop();
        }

        public IEnumerator<T> GetEnumerator()
        {
            var ret = new List<T>();

            while (this.Count() > 0)
            {
                ret.Add(this.Pop());
            }

            foreach (var r in ret)
            {
                this.Push(r);
                yield return r;
            }
        }

        public T[] ToArray()
        {
            T[] array = new T[_sz];
            int i = 0;

            foreach (var r in this)
            {
                array[i++] = r;
            }

            return array;
        }
    }

#if DEBUG
    [TestClass]
    public class Test
    {
        [TestMethod]
        public void Test1()
        {
            var x = 2;
            var y = 2;
            var z = 2;
            var k = 8;
            var a = new List<long> { 4, 6 };
            var b = new List<long> { 1, 5 };
            var c = new List<long> { 3, 8 };
            var exp = new List<long>
            {
                19,
                17,
                15,
                14,
                13,
                12,
                10,
                8,
            };
            var ans = Program.GetAns(k, x, y, z, a, b, c);
            Assert.IsTrue(IsEq(exp, ans));
        }

        [TestMethod]
        public void Test2()
        {
            var x = 3;
            var y = 3;
            var z = 3;
            var k = 5;
            var a = new List<long> { 1, 10, 100 };
            var b = new List<long> { 2, 20, 200 };
            var c = new List<long> { 1, 10, 100 };
            var exp = new List<long>
            {
                400,
                310,
                310,
                301,
                301,
            };
            var ans = Program.GetAns(k, x, y, z, a, b, c);
            Assert.IsTrue(IsEq(exp, ans));
        }

        [TestMethod]
        public void Test_Rnd()
        {
            var rnd = new Random();
            var x = 28;
            var y = 34;
            var z = 25;
            var k = 31;
            var a = Enumerable.Range(0, x).Select(v => ((long)rnd.Next() << 32 | (long)rnd.Next()) + 1).ToList();
            var b = Enumerable.Range(0, y).Select(v => ((long)rnd.Next() << 32 | (long)rnd.Next()) + 1).ToList();
            var c = Enumerable.Range(0, z).Select(v => ((long)rnd.Next() << 32 | (long)rnd.Next()) + 1).ToList();
            var exp = new List<long>();

            for (int i = 0; i < x; i++)
            {
                for (int j = 0; j < y; j++)
                {
                    for (int l = 0; l < z; l++)
                    {
                        exp.Add(a[i] + b[j] + c[l]);
                    }
                }
            }
            exp = exp.OrderByDescending(v => v).Take(k).ToList();

            var ans = Program.GetAns(k, x, y, z, a, b, c);
            Assert.IsTrue(IsEq(exp, ans));
        }

        [TestMethod]
        public void Test_Rnd2()
        {
            var rnd = new Random();
            var x = 31;
            var y = 15;
            var z = 21;
            var k = x * y * z;
            var a = Enumerable.Range(0, x).Select(v => ((long)rnd.Next() << 32 | (long)rnd.Next()) + 1).ToList();
            var b = Enumerable.Range(0, y).Select(v => ((long)rnd.Next() << 32 | (long)rnd.Next()) + 1).ToList();
            var c = Enumerable.Range(0, z).Select(v => ((long)rnd.Next() << 32 | (long)rnd.Next()) + 1).ToList();
            var exp = new List<long>();

            for (int i = 0; i < x; i++)
            {
                for (int j = 0; j < y; j++)
                {
                    for (int l = 0; l < z; l++)
                    {
                        exp.Add(a[i] + b[j] + c[l]);
                    }
                }
            }
            exp = exp.OrderByDescending(v => v).Take(k).ToList();

            var ans = Program.GetAns(k, x, y, z, a, b, c);
            Assert.IsTrue(IsEq(exp, ans));
        }

        [TestMethod]
        public void Test_Rnd3()
        {
            var rnd = new Random();
            var x = 1;
            var y = 3;
            var z = 21;
            var k = 3;
            var a = Enumerable.Range(0, x).Select(v => ((long)rnd.Next() << 32 | (long)rnd.Next()) + 1).ToList();
            var b = Enumerable.Range(0, y).Select(v => ((long)rnd.Next() << 32 | (long)rnd.Next()) + 1).ToList();
            var c = Enumerable.Range(0, z).Select(v => ((long)rnd.Next() << 32 | (long)rnd.Next()) + 1).ToList();
            var exp = new List<long>();

            for (int i = 0; i < x; i++)
            {
                for (int j = 0; j < y; j++)
                {
                    for (int l = 0; l < z; l++)
                    {
                        exp.Add(a[i] + b[j] + c[l]);
                    }
                }
            }
            exp = exp.OrderByDescending(v => v).Take(k).ToList();

            var ans = Program.GetAns(k, x, y, z, a, b, c);
            Assert.IsTrue(IsEq(exp, ans));
        }


        [TestMethod]
        public void Test3()
        {
            var x = 3;
            var y = 3;
            var z = 3;
            var k = 5;
            var a = new List<long> { 10, 10, 10 };
            var b = new List<long> { 0, 0, 0 };
            var c = new List<long> { 0, 0, 0 };
            var exp = new List<long> { 10, 10, 10, 10, 10 };
            var ans = Program.GetAns(k, x, y, z, a, b, c);
            Assert.IsTrue(IsEq(exp, ans));
        }

        bool IsEq(IEnumerable<int> l1, IEnumerable<int> l2)
        {
            if (l1.Count() != l2.Count()) return false;
            for(int i=0; i>l1.Count(); i++)
            {
                if (l1.ElementAt(i) != l2.ElementAt(i)) return false;
            }
            return true;
        }

        bool IsEq(IEnumerable<long> l1, IEnumerable<long> l2)
        {
            if (l1.Count() != l2.Count()) return false;
            for (int i = 0; i > l1.Count(); i++)
            {
                if (l1.ElementAt(i) != l2.ElementAt(i)) return false;
            }
            return true;
        }

        bool IsEq(IEnumerable<double> l1, IEnumerable<double> l2)
        {
            if (l1.Count() != l2.Count()) return false;
            for (int i = 0; i > l1.Count(); i++)
            {
                if (l1.ElementAt(i) != l2.ElementAt(i)) return false;
            }
            return true;
        }

        bool IsEq(double d1, double d2)
        {
            return Math.Abs(d2 - d1) < 1E10;
        }

    }
#endif

    namespace IO
    {
        using System.IO;
        using System.Text;
        using System.Globalization;

        public class Printer : StreamWriter
        {
            static Printer()
            {
                Out = new Printer(Console.OpenStandardOutput()) { AutoFlush = false };
            }

            public static Printer Out { get; set; }

            public override IFormatProvider FormatProvider
            {
                get { return CultureInfo.InvariantCulture; }
            }

            public Printer(Stream stream)
                : base(stream, new UTF8Encoding(false, true))
            {
            }

            public Printer(Stream stream, Encoding encoding)
                : base(stream, encoding)
            {
            }

            public void Write<T>(string format, T[] source)
            {
                base.Write(format, source.OfType<object>().ToArray());
            }

            public void WriteLine<T>(string format, T[] source)
            {
                base.WriteLine(format, source.OfType<object>().ToArray());
            }
        }

        public class StreamScanner
        {
            public StreamScanner(Stream stream)
            {
                str = stream;
            }

            public readonly Stream str;
            private readonly byte[] buf = new byte[1024];
            private int len, ptr;
            public bool isEof;

            public bool IsEndOfStream
            {
                get { return isEof; }
            }

            private byte read()
            {
                if (isEof) return 0;
                if (ptr < len) return buf[ptr++];
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) > 0) return buf[ptr++];
                isEof = true;
                return 0;
            }

            public char Char()
            {
                byte b;
                do b = read(); while ((b < 33 || 126 < b) && !isEof);
                return (char)b;
            }

            public string Scan()
            {
                var sb = new StringBuilder();
                for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                    sb.Append(b);
                return sb.ToString();
            }

            public string ScanLine()
            {
                var sb = new StringBuilder();
                for (var b = Char(); b != '\n'; b = (char)read())
                    if (b == 0) break;
                    else if (b != '\r') sb.Append(b);
                return sb.ToString();
            }

            public long Long()
            {
                if (isEof) return long.MinValue;
                long ret = 0;
                byte b;
                var ng = false;
                do b = read(); while (b != 0 && b != '-' && (b < '0' || '9' < b));
                if (b == 0) return long.MinValue;
                if (b == '-')
                {
                    ng = true;
                    b = read();
                }
                for (; ; b = read())
                {
                    if (b < '0' || '9' < b)
                        return ng ? -ret : ret;
                    ret = ret * 10 + b - '0';
                }
            }

            public int Integer()
            {
                return (isEof) ? int.MinValue : (int)Long();
            }

            public double Double()
            {
                var s = Scan();
                return s != "" ? double.Parse(s, CultureInfo.InvariantCulture) : double.NaN;
            }

            static T[] enumerate<T>(int n, Func<T> f)
            {
                var a = new T[n];
                for (int i = 0; i < n; ++i) a[i] = f();
                return a;
            }

            public char[] Char(int n)
            {
                return enumerate(n, Char);
            }

            public string[] Scan(int n)
            {
                return enumerate(n, Scan);
            }

            public double[] Double(int n)
            {
                return enumerate(n, Double);
            }

            public int[] Integer(int n)
            {
                return enumerate(n, Integer);
            }

            public long[] Long(int n)
            {
                return enumerate(n, Long);
            }
        }
    }
}