Problem:
Score : 300 points

Problem

Takahashi, who is a novice in competitive programming, wants to learn M algorithms.
Initially, his understanding level of each of the M algorithms is 0.

Takahashi is visiting a bookstore, where he finds N books on algorithms.
The i-th book (1\leq i\leq N) is sold for C_i yen (the currency of Japan). If he buys and reads it, his understanding level of the j-th algorithm will increase by A_{i,j} for each j (1\leq j\leq M).
There is no other way to increase the understanding levels of the algorithms.

Takahashi's objective is to make his understanding levels of all the M algorithms X or higher. Determine whether this objective is achievable. If it is achievable, find the minimum amount of money needed to achieve it.

Constraints

All values in input are integers.

1\leq N, M\leq 12

1\leq X\leq 10^5

1\leq C_i \leq 10^5

0\leq A_{i, j} \leq 10^5

Input

Input is given from Standard Input in the following format:

N M X
C_1 A_{1,1} A_{1,2} \cdots A_{1,M}
C_2 A_{2,1} A_{2,2} \cdots A_{2,M}
\vdots
C_N A_{N,1} A_{N,2} \cdots A_{N,M}

Output

If the objective is not achievable, print -1; otherwise, print the minimum amount of money needed to achieve it.

Sample Input 1

3 3 10
60 2 2 4
70 8 7 9
50 2 3 9

Sample Output 1

120

Buying the second and third books makes his understanding levels of all the algorithms 10 or higher, at the minimum cost possible.

Sample Input 2

3 3 10
100 3 1 4
100 1 5 9
100 2 6 5

Sample Output 2

-1

Buying all the books is still not enough to make his understanding levels of all the algorithms 10 or higher.

Sample Input 3

8 5 22
100 3 7 5 3 1
164 4 5 2 7 8
334 7 2 7 2 9
234 4 7 2 8 2
541 5 4 3 3 6
235 4 8 6 9 7
394 3 6 1 6 2
872 8 4 3 7 2

Sample Output 3

1067

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

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;
using static System.Math;
using static AtCoderTemplate.MyConstants;
using static AtCoderTemplate.MyInputOutputs;
using static AtCoderTemplate.MyNumericFunctions;
using static AtCoderTemplate.MyAlgorithm;
using static AtCoderTemplate.MyDataStructure;
using static AtCoderTemplate.MyExtensions;
using static AtCoderTemplate.MyEnumerable;

namespace AtCoderTemplate
{
    public class Program
    {
        public static void Main(string[] args)
        {
            var NMX = ReadInts();
            var N = NMX[0];
            var M = NMX[1];
            var X = NMX[2];
            var C = new int[N];
            var A = new int[N, M];
            foreach (var i in MyEnumerable.Interval(0, N))
            {
                var CA = ReadInts();
                C[i] = CA[0];
                foreach (var j in MyEnumerable.Interval(0, M))
                {
                    A[i, j] = CA[j + 1];
                }
            }
            PrintRow(C);

            var inf = (int)Pow(10, 7);

            var A0 = Enumerable.Repeat(0, M).ToArray();
            var q = new Queue<(int, bool, int, int[])>(new[] { (0, false, 0, A0), (0, true, 0, A0) });
            var ans = inf;

            while (q.Count != 0)
            {
                var (n, b, s, A1) = q.Dequeue();

                if (n == N)
                {
                    if (A1.All(Ai => Ai >= X))
                    {
                        ans = Min(ans, s);
                    }
                    continue;
                }

                var s2 = s;
                var A2 = new List<int>(A1).ToArray();
                if (b)
                {
                    s2 += C[n];
                    foreach (var i in MyEnumerable.Interval(0, M))
                    {
                        A2[i] += A[n, i];
                    }
                }

                q.Enqueue((n + 1, false, s2, A2));
                q.Enqueue((n + 1, true, s2, A2));
            }

            PrintIf(ans != inf, ans, -1);
        }
    }

    public static class MyInputOutputs
    {
        /* Input & Output*/
        public static string Read()
        {
            return Console.ReadLine();
        }

        public static List<string> Reads()
        {
            return Console.ReadLine().Split(' ').ToList();
        }

        public static List<List<string>> ReadRows(int rowNum)
        {
            /*
            入力例
            A1 B1 C1 ... Z1
            A2 B2 C2 ... Z2
            ...
            An Bn Cn ... Zn
           

            出力例
            [[A1, B1, C1, ... Z1], [A2, B2, C2, ... Z2], ... [An, Bn, Cn, ... Zn]]
            */
            return Enumerable.Range(0, rowNum).Select(i => Reads()).ToList();
        }

        public static List<List<string>> ReadColumns(int rowNum, int colNum)
        {
            /*
            入力例
            A1 B1 C1 ... Z1
            A2 B2 C2 ... Z2
            ...
            An Bn Cn ... Zn
           

            出力例
            [[A1, A2, A3, ... An], [B1, B2, B3, ... Bn], ... [Z1, Z2, Z3, ... Zn]]
            */
            var rows = ReadRows(rowNum);
            return Enumerable.Range(0, colNum).Select(i => rows.Select(items => items[i].ToString()).ToList()).ToList();
        }

        public static List<List<string>> ReadGridGraph(int height, int width)
        {
            /*
            入力例
            A1B1C1...Z1
            A2B2C2...Z2
            ...
            AnBnCn...Zn
           

            出力例
            [[A1, B1, C1, ... Z1], [A2, B2, C2, ... Z2], ... [An, Bn, Cn, ... Zn]]
            */
            return Enumerable.Range(0, height)
                .Select(i =>
                   Read()
                   .Select(c => c.ToString())
                   .ToList()
                ).ToList();
        }

        public static int ToInt(this string str)
        {
            return int.Parse(str);
        }

        public static long ToLong(this string str)
        {
            return long.Parse(str);
        }

        public static List<int> ToInts(this List<string> strs)
        {
            return strs.Select(str => str.ToInt()).ToList();
        }

        public static List<long> ToLongs(this List<string> strs)
        {
            return strs.Select(str => str.ToLong()).ToList();
        }

        public static int ReadInt()
        {
            return Read().ToInt();
        }
        public static long ReadLong()
        {
            return Read().ToLong();
        }

        public static List<int> ReadInts()
        {
            return Reads().ToInts();
        }
        public static List<long> ReadLongs()
        {
            return Reads().ToLongs();
        }

        public static void Print<T>(T item)
        {
            Console.WriteLine(item);
        }
        public static void PrintIf<T1, T2>(bool condition, T1 trueResult, T2 falseResult)
        {
            if (condition)
            {
                Console.WriteLine(trueResult);
            }
            else
            {
                Console.WriteLine(falseResult);
            }
        }

        public static void PrintRow<T>(IEnumerable<T> list)
        {
            /* 横ベクトルで表示
            A B C D ...
            */
            if (!list.IsEmpty())
            {
                Console.Write(list.First());
                foreach (var item in list.Skip(1))
                {
                    Console.Write($" {item}");
                }
            }
            Console.Write("\n");
        }
        public static void PrintColomn<T>(IEnumerable<T> list)
        {
            /* 縦ベクトルで表示
            A
            B
            C
            D
            ...
            */
            foreach (var item in list)
            {
                Console.WriteLine(item);
            }
        }
        public static void PrintRows<T>(IEnumerable<IEnumerable<T>> sources)
        {
            foreach (var row in sources)
            {
                PrintRow(row);
            }
        }

        public static void PrintGridGraph<T>(IEnumerable<IEnumerable<T>> sources)
        {
            foreach (var row in sources)
            {
                Print(String.Concat(row));
            }
        }
    }

    public static class MyConstants
    {
        public static IEnumerable<char> lowerAlphabets = Enumerable.Range('a', 'z' - 'a' + 1).Select(i => (char)i);
        public static IEnumerable<char> upperAlphabets = Enumerable.Range('A', 'Z' - 'A' + 1).Select(i => (char)i);

        public static int p1000000007 = (int)Pow(10, 9) + 7;
    }

    public static class MyNumericFunctions
    {

        public static bool IsEven(int a)
        {
            return a % 2 == 0;
        }
        public static bool IsOdd(int a)
        {
            return !IsEven(a);
        }
        public static bool IsEven(long a)
        {
            return a % 2L == 0L;
        }
        public static bool IsOdd(long a)
        {
            return !IsEven(a);
        }

        public static int[,] PascalsTriangle(int nmax, int kmax, int divisor)
        {
            var comb = new int[2000 + 1, 2000 + 1];
            foreach (var n in MyEnumerable.Interval(0, 2000 + 1))
            {
                foreach (var k in MyEnumerable.Interval(0, 2000 + 1))
                {
                    if (n < k) continue;

                    if (k == 0)
                    {
                        comb[n, k] = 1;
                    }
                    else
                    {
                        comb[n, k] = (int)(((long)comb[n - 1, k - 1] + comb[n - 1, k]) % divisor);
                    }
                }
            }

            return comb;
        }
        /// <summary>
        /// Mod計算
        /// </summary>
        public class Mods
        {
            int divisor;
            public Mods(int divisor)
            {
                this.divisor = divisor;
            }
            public int Mod(long a)
            {
                var b = (int)(a % divisor);
                if (b < 0)
                {
                    return b + divisor;
                }
                else
                {
                    return b;
                }
            }

            public int Add(int a, int b)
            {
                return Mod(Mod(a) + Mod(b));
            }
            public int Sub(int a, int b)
            {
                return Mod(Mod(a) - Mod(b));
            }

            public int Mul(int a, int b)
            {
                return Mod((long)Mod(a) * Mod(b));
            }

            public int Pow(int b, int n)
            {
                var digit = (int)Math.Log(n, 2.0);
                var pows = Interval(0, digit + 1)
                    .Scanl(b, (accm, _) => Mul(accm, accm))
                    .ToArray();
                return Interval(0, digit + 1)
                    .Aggregate(1, (accm, i) => ((n >> i) & 1) == 1 ? Mul(accm, pows[i]) : accm);
            }
            public int Inv(int a)
            {
                return Pow(a, divisor - 2);
            }
            public int Div(int a, int b)
            {
                return Mul(a, Inv(b));
            }

            public int Perm(int n, int k)
            {
                if (n < 0 || k < 0) throw new ArgumentOutOfRangeException();

                if (n < k)
                {
                    return 0;
                }
                else
                {
                    return Interval(n - k + 1, n + 1)
                        .Aggregate(1, Mul);
                }
            }

            public List<int> FactTable(int nMax)
            {
                return Interval(1, nMax + 1)
                    .Scanl(1, Mul)
                    .ToList();
            }

            public List<List<int>> CombTable(int nMax)
            {
                var table = Enumerable.Repeat(0, nMax + 1)
                    .Select(_ =>
                       Enumerable.Repeat(0, nMax + 1).ToList()
                    ).ToList();

                foreach (var n in Interval(0, nMax + 1))
                {
                    foreach (var k in Interval(0, nMax + 1))
                    {
                        if (n < k)
                        {
                            table[n][k] = 0;
                        }
                        else if (k == 0)
                        {
                            table[n][k] = 1;
                        }
                        else
                        {
                            table[n][k] = Add(table[n - 1][k - 1], table[n - 1][k]);
                        }
                    }
                }
                return table;
            }
        }

        /// <summary>
        /// 最大公約数を得る 
        /// O(log N)
        /// </summary>
        /// <param name="m">自然数</param>
        /// <param name="n">自然数</param>
        /// <returns></returns>
        public static long GCD(long m, long n)
        {
            // GCD(m,n) = GCD(n, m%n)を利用
            // m%n = 0のとき、mはnで割り切れるので、nが最大公約数
            if (m <= 0L || n <= 0L) throw new ArgumentOutOfRangeException();

            if (m < n) return GCD(n, m);
            while (m % n != 0L)
            {
                var n2 = m % n;
                m = n;
                n = n2;
            }
            return n;
        }

        /// <summary>
        /// 最小公倍数を得る
        /// O(log N)
        /// </summary>
        /// <param name="m"></param>
        /// <param name="n"></param>
        /// <returns></returns>
        public static long LCM(long m, long n)
        {
            var ans = checked((long)(BigInteger.Multiply(m, n) / GCD(m, n)));
            return ans;
        }

        /// <summary>
        /// 約数列挙(非順序)
        /// O(√N)
        /// </summary>
        /// <param name="m">m > 0</param>
        /// <returns></returns>
        public static IEnumerable<long> Divisor(long m)
        {
            if (!(m > 0)) throw new ArgumentOutOfRangeException();

            var front = Enumerable.Range(1, (int)Sqrt(m))
                .Select(i => (long)i)
                .Where(d => m % d == 0);
            return front.Concat(front.Where(x => x * x != m).Select(x => m / x));
        }

        public static IEnumerable<int> Divisor(int m)
        {
            if (!(m > 0)) throw new ArgumentOutOfRangeException();

            var front = Enumerable.Range(1, (int)Sqrt(m))
                .Where(d => m % d == 0);
            return front.Concat(front.Where(x => x * x != m).Select(x => m / x));
        }

        /// <summary>
        /// 公約数列挙(非順序)
        /// O(√N)
        /// </summary>
        /// <param name="m">m > 0</param>
        /// <param name="n">n > 0 </param>
        /// <returns></returns>
        public static IEnumerable<long> CommonDivisor(long m, long n)
        {
            if (!(m > 0)) throw new ArgumentOutOfRangeException();

            if (m < n) return CommonDivisor(n, m);
            return Divisor(m).Where(md => n % md == 0);
        }

        public static IEnumerable<int> CommonDivisor(int m, int n)
        {
            if (!(m > 0)) throw new ArgumentOutOfRangeException();

            if (m < n) return CommonDivisor(n, m);
            return Divisor(m).Where(md => n % md == 0);
        }

        /// <summary>
        /// エラトステネスの篩 O(N loglog N)
        /// </summary>
        /// <param name="n"></param>
        /// <returns></returns>
        public static IEnumerable<int> Primes(int n)
        {
            if (!(n > 1)) throw new ArgumentOutOfRangeException();

            var ps = Interval(2, n + 1);
            while (!ps.IsEmpty() && ps.First() <= Sqrt(n))
            {
                var m = ps.First();
                ps = ps.Where(p => p % m != 0);
            }
            return ps;
        }

        /// <summary>
        /// 素因数分解 O(N loglog N)
        /// </summary>
        /// <param name="n"></param>
        /// <returns></returns>
        public static IEnumerable<int> PrimeFactrization(int n)
        {
            if (!(n > 1)) throw new ArgumentOutOfRangeException();

            var e = new int[n + 1];
            var p = n;
            var ps = Primes(n).ToList();
            var i = 0;
            while (p != 1)
            {
                if (p % ps[i] == 0)
                {
                    e[ps[i]] += 1;
                    p /= ps[i];
                    continue;
                }
                i += 1;
            }
            return e;
        }

        /// <summary>
        /// 順列を得る
        /// O(N!)
        /// </summary>
        public static IEnumerable<IEnumerable<T>> Perms<T>(IEnumerable<T> source, int n)
        {
            if (n == 0 || source.IsEmpty() || source.Count() < n)
            {
                return Enumerable.Empty<IEnumerable<T>>();
            }
            else if (n == 1)
            {
                return source.Select(i => new List<T> { i });
            }
            else
            {
                var nexts = source.Select((x, i) =>
                   new { next = source.Take(i).Concat(source.Skip(i + 1)), selected = source.Take(i + 1).Last() });
                return nexts.SelectMany(next => Perms(next.next, n - 1).Select(item => item.Prepend(next.selected)));
            }
        }
    }

    public static class MyAlgorithm
    {
        /// <summary>
        /// めぐる式二分探索法
        /// O(log N)
        /// </summary>
        /// <param name="list">探索するリスト</param>
        /// <param name="predicate">条件の述語関数</param>
        /// <param name="ng">条件を満たさない既知のindex</param>
        /// <param name="ok">条件を満たす既知のindex</param>
        /// <returns>条件を満たすindexの内、隣がfalseとなるtrueのindexを返す</returns>
        public static long BinarySearch<T>(Func<long, bool> predicate, long ng, long ok)
        {
            while (Abs(ok - ng) > 1)
            {
                var mid = (ok + ng) / 2L;
                if (predicate(mid))
                {
                    ok = mid;
                }
                else
                {
                    ng = mid;
                }
            }
            return ok;
        }

        /// <summary>
        /// ランレングス符号化
        /// </summary>
        /// <param name="source">元の文字列</param>
        /// <returns>連続した文字をひとまとめにし、その文字と長さのペアの列を得る</returns>
        /// <example>RunLengthEncoding("aaabccdddd") => [(a,3), (b,1), (c,2), (d,4)]</example>
        public static IEnumerable<Tuple<string, int>> RunLengthEncoding(string source)
        {
            var cutIndexes = Interval(1, source.Length)
                .Where(i => source[i] != source[i - 1])
                .Prepend(0)
                .Append(source.Length);
            return cutIndexes
                .MapAdjacent((i0, i1) => Tuple.Create<string, int>(source[i0].ToString(), i1 - i0));
        }

        /// <summary>
        /// 3分探索法 O(log N)
        /// </summary>
        /// <param name="l">定義域の最小値</param>
        /// <param name="r">定義域の最大値</param>
        /// <param name="f">凸な関数</param>
        /// <param name="allowableError">許容誤差</param>
        /// <param name="isDownwardConvex">下に凸か（falseならば上に凸）</param>
        /// <returns>凸関数f(x)の許容誤差を含む極値への元を得る</returns>
        public static double TernarySerch(double l, double r, Func<double, double> f, double allowableError, bool isDownwardConvex)
        {
            while (r - l >= allowableError)
            {
                var ml = l + (r - l) / 3; // mid left
                var mr = l + (r - l) / 3 * 2.0; // mid right
                var fml = f(ml);
                var fmr = f(mr);
                if (isDownwardConvex)
                {
                    if (fml < fmr)
                    {
                        r = mr;
                    }
                    else if (fml > fmr)
                    {
                        l = ml;
                    }
                    else
                    {
                        l = ml;
                        r = mr;
                    }
                }
                else
                {
                    if (fml < fmr)
                    {
                        l = ml;
                    }
                    else if (fml > fmr)
                    {
                        r = mr;
                    }
                    else
                    {
                        l = ml;
                        r = mr;
                    }
                }
            }
            return l;
        }

        /// <summary>
        /// フラグが立っているか判定する
        /// </summary>
        /// <param name="flags">フラグ</param>
        /// <param name="digit">判定する桁目</param>
        /// <returns>digit のフラグが立っていれば true</returns>
        public static bool IsFlag(long flags, int digit)
        {
            return ((flags >> digit) & 1L) == 1L;
        }
    }

    public static class MyDataStructure
    {
        public class UnionFind
        {
            List<int> parent;
            List<int> size;
            public UnionFind(int N)
            {
                // 最初はすべて異なるグループ番号(root)が割り当てられる
                parent = Enumerable.Range(0, N).ToList();
                size = Enumerable.Repeat(1, N).ToList();
            }

            // 頂点uの属するグループ番号(root)を探す
            int Root(int u)
            {
                if (parent[u] == u)
                {
                    return u;
                }
                else
                {
                    var root = Root(parent[u]);
                    parent[u] = root; // 経路圧縮
                    return root;
                }
            }

            // 2つのグループを統合する(rootが異なる場合、同じrootにする)
            public void Unite(int u, int v)
            {
                int root_u = Root(u);
                int root_v = Root(v);
                if (root_u == root_v) return;
                parent[root_u] = root_v; // root_vをroot_uの親とする
                size[root_v] += size[root_u];
            }

            public int Size(int u)
            {
                return size[Root(u)];
            }

            public bool IsSame(int u, int v)
            {
                int root_u = Root(u);
                int root_v = Root(v);
                return root_u == root_v;
            }
        }

        /// <summary>
        /// Comparison<T>(順序を定める高階関数 t -> t -> Orderingみたいなもの)からIComparer<T>の実体へ変換するクラス
        /// </summary>
        /// <typeparam name="T">順序付けられる型</typeparam>
        public class ComparisonToComparerConverter<T> : IComparer<T>
        {
            Comparison<T> comparison;
            public ComparisonToComparerConverter(Comparison<T> comparison)
            {
                this.comparison = comparison;
            }
            public int Compare(T x, T y)
            {
                return comparison(x, y);
            }
        }

        /// <summary>
        /// IComparble<T>を持つ型Tから、その逆順序であるComparison<T>を得る
        /// </summary>
        /// <typeparam name="T">順序付きの型</typeparam>
        /// <returns>IComparable<T>の逆順序</returns>
        public static Comparison<T> ReverseOrder<T>() where T : IComparable<T>
        {
            return (x, y) => -x.CompareTo(y);
        }

        /// <summary>
        /// 優先度付きキュー(デフォルトでは昇順)
        /// 先頭参照・要素数がO(1)、要素の追加・先頭削除がO(log N)
        /// </summary>
        /// <typeparam name="T">順序付きの型</typeparam>
        public class PriorityQueue<T> : IEnumerable<IEnumerable<T>>
            where T : IComparable<T>
        {
            SortedDictionary<T, Queue<T>> dict;
            int size = 0;

            public PriorityQueue()
            {
                dict = new SortedDictionary<T, Queue<T>>();
            }

            public PriorityQueue(IComparer<T> comparer)
            {
                dict = new SortedDictionary<T, Queue<T>>(comparer);
            }

            public PriorityQueue(Comparison<T> comparison)
            {
                dict = new SortedDictionary<T, Queue<T>>(new ComparisonToComparerConverter<T>(comparison));
            }

            public void Enqueue(T item)
            {
                if (dict.ContainsKey(item))
                {
                    dict[item].Enqueue(item);
                }
                else
                {
                    var added = new Queue<T>();
                    added.Enqueue(item);
                    dict.Add(item, added);
                }
                size += 1;
            }

            public T Peek()
            {
                return dict.First().Value.First();
            }

            public int Size()
            {
                return size;
            }

            public T Dequeue()
            {
                var first = dict.First();
                if (first.Value.Count <= 1)
                {
                    dict.Remove(first.Key);
                }
                size -= 1;
                return first.Value.Dequeue();
            }

            public IEnumerator<IEnumerable<T>> GetEnumerator()
            {
                foreach (var kv in dict)
                {
                    yield return kv.Value;
                }
            }

            IEnumerator IEnumerable.GetEnumerator()
            {
                return GetEnumerator();
            }
        }
    }

    public static class MyExtensions
    {
        public static bool IsEmpty<T>(this IEnumerable<T> source)
        {
            return !source.Any();
        }

        /// <summary>
        /// シーケンスの隣り合う要素を2引数の関数に適用したシーケンスを得る
        /// </summary>
        /// <para>O(N)</para>
        /// <param name="source">元のシーケンス</param>
        /// <param name="func">2引数関数</param>
        /// <example>[1,2,3,4].MapAdjacent(f) => [f(1,2), f(2,3), f(3,4)]</example>
        public static IEnumerable<TR> MapAdjacent<T1, TR>(this IEnumerable<T1> source, Func<T1, T1, TR> func)
        {
            var list = source.ToList();
            return Enumerable.Range(1, list.Count - 1)
                .Select(i => func(list[i - 1], list[i]));
        }

        /// <summary>
        /// 累積項を要素にもつシーケンスを得る(初項は、first)
        /// <para>O(N)</para>
        /// </summary>
        /// <param name="source">元のシーケンス</param>
        /// <param name="func">2引数関数f</param>
        /// <param name="first">func(first, source[0])のための初項</param>
        /// <example> [1,2,3].Scanl1(0,f) => [0, f(0,1), f(f(0,1),2), f(f(f(0,1),2),3)]</example>
        public static IEnumerable<TR> Scanl<T, TR>(this IEnumerable<T> source, TR first, Func<TR, T, TR> func)
        {
            var list = source.ToList();
            var result = new List<TR> { first };
            foreach (var i in Enumerable.Range(0, source.Count()))
            {
                result.Add(func(result[i], list[i]));
            }
            return result;
        }
        /// <summary>
        /// 累積項を要素にもつシーケンスを得る（初項は、source.First()）
        /// <para>O(N)</para>
        /// </summary>
        /// <param name="source">要素数1以上のシーケンス</param>
        /// <param name="func">2引数関数f</param>
        /// <example> [1,2,3].Scanl1(f) => [1, f(1,2), f(f(1,2),3)]</example>
        public static IEnumerable<T> Scanl1<T>(this IEnumerable<T> source, Func<T, T, T> func)
        {
            if (source.IsEmpty()) throw new ArgumentOutOfRangeException();
            var list = source.ToList();
            var result = new List<T> { list[0] };
            foreach (var i in Enumerable.Range(1, source.Count() - 1))
            {
                result.Add(func(result[i - 1], list[i]));
            }
            return result;
        }

        /// <summary>
        /// 数列の和をdivisorで割った余りを計算する
        /// </summary>
        /// <param name="source">数列</param>
        /// <param name="divisor">割る数</param>
        /// <returns>数列の和をdivisorで割った余り</returns>
        public static int Sum(this IEnumerable<int> source, int divisor)
        {
            return source.Aggregate(0, (a, b) => (int)(((long)a + b) % divisor));
        }

        /// <summary>
        /// 数列の積を計算する
        /// </summary>
        /// <param name="source">数列</param>
        /// <returns>数列の積</returns>
        public static long Product(this IEnumerable<long> source)
        {
            return source.Aggregate(1L, (a, b) => a * b);
        }

        /// <summary>
        /// 数列の積をdivisorで割った余りを計算する
        /// </summary>
        /// <param name="source">数列</param>
        /// <param name="divisor">割る数</param>
        /// <returns>数列の積をdivisorで割った余り</returns>
        public static int Product(this IEnumerable<int> source, int divisor)
        {
            return source.Aggregate(1, (a, b) => (int)(((long)a * b) % divisor));
        }

    }

    public static class MyEnumerable
    {
        /// <summary>
        /// 左閉右開区間 [startIndex,endIndex) を得る
        /// </summary>
        /// <param name="startIndex">始まりのインデックス。含む</param>
        /// <param name="endIndex">終わりのインデックス。含まない</param>
        public static IEnumerable<int> Interval(int startIndex, int endIndex)
        {
            if (endIndex - startIndex < 0) new ArgumentException();
            return Enumerable.Range(startIndex, endIndex - startIndex);
        }

        /// <summary>
        /// フラグから分割するindexの位置の列への変換
        /// </summary>
        /// <param name="flags">二進数によるフラグ</param>
        /// <param name="flagSize">フラグの数</param>
        /// <returns>分割するindexの位置の列</returns>
        /// <example> CutFlagToCutIndex(10110) => [0, 2, 3, 5, 6]</example>
        public static IEnumerable<int> CutFlagToIndexes(int flags)
        {
            int flagSize = (int)Log(flags, 2);
            var indexes = new List<int> { 0 };
            foreach (var i in MyEnumerable.Interval(0, flagSize))
            {
                if ((flags >> i) % 2 == 1)
                {
                    indexes.Add(i + 1);
                }
            }
            indexes.Add(flagSize + 1);
            return indexes;
        }
    }
    public static class Template
    {
        public static void TreeBFS()
        {
            // 与えられた頂点数と辺
            var N = ReadInt();
            var ab = ReadColumns(N, 2);
            var a = ab[0].ToInts();
            var b = ab[1].ToInts();

            // 木構造の初期化
            var tree = MyEnumerable.Interval(0, N)
            .Select(i => new List<int>())
            .ToList();

            foreach (var i in MyEnumerable.Interval(0, N - 1))
            {
                tree[a[i] - 1].Add(b[i] - 1);
                tree[b[i] - 1].Add(a[i] - 1);
            }

            var order = new Queue<int>();   // BFS
            // var order = new Stack<int>();   // DFS

            var reached = Enumerable.Repeat(false, N)
            .ToArray();

            // 最初の頂点を追加
            order.Enqueue(0);
            reached[0] = true;

            // 求めたい値
            var ans = false;

            // 実行
            while (!order.IsEmpty())
            {
                var node = order.Dequeue(); // 現在見ている頂点
                Func<int, bool> pruningCondition = child => false; // 枝刈り条件
                var nexts = tree[node]
                .Where(child => !reached[child] && !pruningCondition(child));

                // ************** 処理 *******************

                // 現在の頂点による処理

                // 現在の頂点と、次の頂点による処理
                foreach (var next in nexts)
                {

                }

                // ***************************************

                foreach (var next in nexts)
                {
                    order.Enqueue(next);
                    reached[next] = true;
                }
            }
        }

        public static void GridGraphBFS()
        {
            // 与えられた高さと幅のグリッドグラフ
            var HW = ReadInts();
            var H = HW[0];
            var W = HW[1];
            var S = ReadGridGraph(H, W);

            var order = new Queue<(int, int)>();
            var reached = new bool[H, W];
            foreach (var h in MyEnumerable.Interval(0, H))
            {
                foreach (var w in MyEnumerable.Interval(0, W))
                {
                    reached[h, w] = false;
                }
            }

            // 最初の頂点を追加
            order.Enqueue((0, 0));
            reached[0, 0] = true;

            // 求めたい値
            var ans = false;

            // 実行
            while (!order.IsEmpty())
            {
                var (y0, x0) = order.Dequeue();

                // 周りのマス
                var allSides = new[]{
                    (y0, x0 -1 ),   // left
                    (y0 - 1, x0),   // up
                    (y0, x0 + 1),   // right
                    (y0 + 1, x0),   // down
                }
                .Where(t =>
                {
                    var (y, x) = t;
                    return 0 <= y && y < H && 0 <= x && x < W;
                });

                // 枝刈り条件
                Func<int, int, bool> pruningCondition = (y, x) => false;

                var nexts = allSides
                    .Where(t =>
                    {
                        var (y, x) = t;
                        return !reached[y, x] && !pruningCondition(y, x);
                    });

                // **************** 処理 ******************

                // 現在の頂点による処理

                // 現在の頂点と、次の頂点による処理
                foreach (var (y, x) in nexts)
                {

                }

                // ****************************************

                foreach (var next in nexts)
                {
                    var (y, x) = next;
                    order.Enqueue(next);
                    reached[y, x] = true;
                }
            }
        }

        public static void WarshallFloyd()
        {
            // グラフの入力
            var NM = ReadInts();
            var N = NM[0];
            var M = NM[1];
            var abc = ReadColumns(M, 3);
            var a = abc[0].ToInts();
            var b = abc[1].ToInts();
            var c = abc[2].ToInts();

            // 隣接行列の初期化
            var inf = (long)Pow(2L, 60);
            var G = new long[N, N];
            foreach (var h in MyEnumerable.Interval(0, N))
            {
                foreach (var w in MyEnumerable.Interval(0, N))
                {
                    if (h == w)
                    {
                        G[h, w] = 0L;
                    }
                    else
                    {
                        G[h, w] = inf;
                    }
                }
            }
            foreach (var i in MyEnumerable.Interval(0, M))
            {
                G[a[i], b[i]] = c[i];
                G[b[i], a[i]] = c[i];
            }

            // 実行
            foreach (var k in MyEnumerable.Interval(0, N))
            {
                foreach (var i in MyEnumerable.Interval(0, N))
                {
                    foreach (var j in MyEnumerable.Interval(0, N))
                    {
                        G[i, j] = Min(G[i, j], G[i, k] + G[k, j]);
                    }
                }
            }
        }
    }
}