Problem:
Score : 300 points

Problem Statement

We have a grid of H rows and W columns of squares. The color of the square at the i-th row from the top and the j-th column from the left (1 \leq i \leq H, 1 \leq j \leq W) is given to you as a character c_{i,j}: the square is white if c_{i,j} is ., and black if c_{i,j} is #.

Consider doing the following operation:

Choose some number of rows (possibly zero), and some number of columns (possibly zero). Then, paint red all squares in the chosen rows and all squares in the chosen columns.

You are given a positive integer K. How many choices of rows and columns result in exactly K black squares remaining after the operation? Here, we consider two choices different when there is a row or column chosen in only one of those choices.

Constraints

1 \leq H, W \leq 6

1 \leq K \leq HW

c_{i,j} is . or #.

Input

Input is given from Standard Input in the following format:

H W K
c_{1,1}c_{1,2}...c_{1,W}
c_{2,1}c_{2,2}...c_{2,W}
:
c_{H,1}c_{H,2}...c_{H,W}

Output

Print an integer representing the number of choices of rows and columns satisfying the condition.

Sample Input 1

2 3 2
..#
###

Sample Output 1

5

Five choices below satisfy the condition.

The 1-st row and 1-st column

The 1-st row and 2-nd column

The 1-st row and 3-rd column

The 1-st and 2-nd column

The 3-rd column

Sample Input 2

2 3 4
..#
###

Sample Output 2

1

One choice, which is choosing nothing, satisfies the condition.

Sample Input 3

2 2 3
##
##

Sample Output 3

0

Sample Input 4

6 6 8
..##..
.#..#.
#....#
######
#....#
#....#

Sample Output 4

208

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

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Text;
using System.Threading.Tasks;
using System.Globalization;
using System.Collections;

namespace CompetitiveProgramming
{
    public class CPair<T, U> : IComparable where T : IComparable<T>
    {
        public readonly T Item1;
        public readonly U Item2;

        public int CompareTo(object obj)
        {
            CPair<T, U> castedObj = (CPair<T, U>)obj;
            return this.Item1.CompareTo(castedObj.Item1);
        }

        public CPair(T t, U u)
        {
            Item1 = t;
            Item2 = u;
        }
    }
    class Pair : IComparable
    {
        public long x;
        public long a;
        public long b;
        public long c;
        public Pair(long X, long Y, long Z, long D)
        {
            x = X; a = Y; b = Z;
            c = D;
        }
        public Pair(long a,long x)
        {
            this.a = a;this.x = x;
        }
        public Pair(long a, long b,long c)
        {
            this.a = a; this.b = b;this.c = c;
        }

        public int CompareTo(object obj)
        {
            Pair kore = (Pair)obj;
            return x.CompareTo(kore.x);
        }
        public static bool operator <=(Pair p1, Pair p2)
        {
            if (p1.x != p2.x) { return p1.x <= p2.x; }
            else { return p1.a <= p2.a; }
        }
        public static bool operator >=(Pair p1, Pair p2)
        {
            if (p1.x != p2.x) { return p1.x >= p2.x; }
            else { return p1.a >= p2.a; }
        }
    }

    class Program
    {
        static bool[] visited;
        static List<int>[] togo;

        static void Main(string[] args)
        {
        //    Utils.BinarySearch.Search
            long mod = 1000000007;
            StreamScanner scan = new StreamScanner(System.Console.OpenStandardInput());;
            long h = scan.Long();
            long w = scan.Long();
            long k = scan.Long();
            char[,] c = scan.Char(h, w);
            long first = 0;
            for (long i = 0; i < h; i++)
            {
                for (long j = 0; j < w; j++)
                {
                    if (c[i, j] == '#')
                        first++;
                }
            }
            long ans = 0;
            
            for (long i = 0; i < 1 << (int)h; i++)
            {
                for (long j = 0; j < 1 << (int)w; j++)
                {
                    long buf = first;
                    for (int n = 0; n < h; n++)
                    {
                        if(Bit(i,n))
                        {
                            for (long b = 0; b < w; b++)
                            {
                                if (c[n, b] == '#')
                                    buf--;
                            }
                        }
                    }
                    for (int m = 0; m < w; m++)
                    {
                        if(Bit(j,m))
                        {
                            for(int b = 0;b<h;b++)
                            {
                                if (!Bit(i, b) && c[b, m] == '#')
                                    buf--;
                            }
                        }
                    }
                    if (buf == k)
                        ans++;
                }
                
            }
            Console.WriteLine(ans);
        }
       static bool isOK(long mid, long[] a, long[] f,long k)
        {
            long count = 0;
            for (long i = 0; i < a.Length; i++)
            {
                long togo = mid / f[i]; ;
                count += Math.Max(0, a[i] - togo);
            }
            return (count<= k);
        }


        static bool DFS(int now, int last)
        {
            visited[now] = true;
            for (int i = 0; i < togo[now].Count; i++)
            {
                if (togo[now][i] == last) { continue; }
                if (visited[togo[now][i]]) { return false; }
                if (!DFS(togo[now][i], now)) { return false; }
            }
            return true;
        }



        static int binary_search<T>(T key, T[] moto, Func<int, T, T[], bool> isok)
        {
            int ng = -1;
            int ok = moto.Length;
            while (Math.Abs(ok - ng) > 1)
            {
                int mid = (ok + ng) / 2;
                if (isok(mid, key, moto)) { ok = mid; }
                else { ng = mid; }
            }
            return ok;
        }
        static int binary_search(int key, int[] moto)
        {
            int ng = -1;
            int ok = moto.Length;
            while (Math.Abs(ok - ng) > 1)
            {
                int mid = (ok + ng) / 2;
                if (isOK(mid, key, moto)) { ok = mid; }
                else { ng = mid; }
            }
            return ok;
        }
        static int[] Togo;
        static bool CheckBipartite(int now, int prev, List<int>[] g, int c = 0)//0,-1が想定
        {
            if (Togo == null) Togo = Enumerable.Repeat(-1, g.Length).ToArray();
            foreach (var next in g[now])
            {
                if (next == prev) continue;
                var nextc = Togo[next];
                if (nextc >= 0 && nextc == c) return false;
                if (nextc >= 0) continue;
                Togo[next] = (c + 1) % 2;
                var res = CheckBipartite(next, now, g, (c + 1) % 2);
                if (res == false) return false;
            }
            return true;
        }
        static bool isOK(int index, int key, int[] moto)
        {
            return moto[index] > key;
        }
        public static int bitCount(int k)
        {
            uint i = (uint)k;
            i = i - ((i >> 1) & 0x55555555);
            i = (i & 0x33333333) + ((i >> 2) & 0x33333333);
            i = (i + (i >> 4)) & 0x0f0f0f0f;
            i = i + (i >> 8);
            i = i + (i >> 16);
            return (int)i & 0x3f;
        }
        static int check(long moto, long tai)
        {
            if (tai % moto != 0) { return 0; }
            while (tai > 1)
            {
                if (tai % moto == 0) { tai /= moto; }
                else if (tai > moto) { tai %= moto; }
                else { tai -= moto; }
            }
            if (tai == 1) { return 1; }
            else { return 0; }
        }

        static bool NextPermutation(int[] array)
        {
            var size = array.Length;
            var ok = false;

            //array[i]<array[i+1]を満たす最大のiを求める
            int i = size - 2;
            for (; 0 <= i; i--)
            {
                if (array[i] < array[i + 1])
                {
                    ok = true;
                    break;
                }
            }

            //全ての要素が降順の場合、NextPermutationは存在しない
            if (ok == false) return false;

            //array[i]<array[j]を満たす最大のjを求める
            int j = size - 1;
            for (; ; j--)
            {
                if (array[i] < array[j]) break;
            }

            //iとjの要素をswapする
            var tmp = array[i];
            array[i] = array[j];
            array[j] = tmp;

            //i以降の要素を反転させる
            Array.Reverse(array, i + 1, size - (i + 1));

            return true;
        }
        static List<int[]> GetAllPermutation(int N)
        {
            var res = new List<int[]>();
            var array = Enumerable.Range(0, N).ToArray();
            do
            {
                var copy = new int[N];
                array.CopyTo(copy, 0);
                res.Add(copy);
            } while (NextPermutation(array));

            return res;
        }
        static List<int[]> GetAllPermutation(int[] array)
        {
            var size = array.Length;
            var res = new List<int[]>();
            do
            {
                var copy = new int[size];
                array.CopyTo(copy, 0);
                res.Add(copy);
            } while (NextPermutation(array));

            return res;
        }

        static long Syakutori(int[] a, long k)//しゃくとり法モデル:連続部分列に含まれるすべての要素の値の和がK以上
        {
            long sum = 0;
            int right = 0;
            long ans = 0;
            for (int left = 0; left < a.Length; left++)
            {
                while (right < a.Length && sum < k)
                {
                    sum += a[right]; right++;

                }
                if (k <= sum) { ans += a.Length - right + 1; }
                if (left == right) { right++; }
                else { sum -= a[left]; }
            }
            return ans;
        }
        static int[] zalgo(String s)
        {
            int n = s.Length;
            int[] ret = new int[n];
            ret[0] = n;
            int i = 1, j = 0;
            while (i < n)
            {
                while (i + j < n && s[j] == s[i + j])
                    ++j;
                ret[i] = j;
                if (j == 0)
                {
                    ++i;
                    continue;
                }
                int k = 1;
                while (i + k < n && k + ret[k] < j)
                {
                    ret[i + k] = ret[k];
                    ++k;
                }
                i += k;
                j -= k;
            }
            return ret;
        }
        static int gcd(int a, int b)
        {
            if (a % b == 0) { return b; }
            return gcd(b, a % b);
        }
        static long gcd(long a, long b)
        {
            if (a % b == 0) { return b; }
            return gcd(b, a % b);
        }
        public static List<long> factorialMemo = new List<long>() { 1 };
        static long Fact(int x)
        {
            for (int i = factorialMemo.Count; i <= x; i++) factorialMemo.Add(factorialMemo.Last() * i);
            return factorialMemo[x];
        }
        static int[] yousogotoplus(int[] a, int[] b)
        {
            int[] c = new int[a.Length];
            for (int i = 0; i < a.Length; i++)
            {
                c[i] = a[i] + b[i];
            }
            return c;
        }
        static int[] yousogotoplus(int[] a, int[,] b, int l)
        {
            int[] c = new int[a.Length];
            for (int i = 0; i < a.Length; i++)
            {
                c[i] = a[i] + b[l, i];
            }
            return c;
        }

        static int[] yousogotoplus(int[,] a, int l)
        {
            int[] c = new int[a.GetLength(1)];
            for (int i = 0; i < a.GetLength(1); i++)
            {
                c[i] = a[l, i];
            }
            return c;
        }

        public static long LowerBound(long[] a, long key)
        {
            long l = 0;
            long r = a.Length;
            long mid;
            while (l < r)
            {
                mid = ((r - l) / 2) + l;
                if (a[mid] <= key)
                    l = mid + 1;
                else
                    r = mid;
            }
            return l;
        }
        public static long LIS(long[] a)
        {
            var dp = Enumerable.Range(0, a.Length).Select(x => long.MaxValue).ToArray();
            for (int i = 0; i < a.Length; i++)
            {
                dp[LowerBound(dp, a[i])] = a[i];
            }
            return LowerBound(dp, long.MaxValue - 1);
        }

        static long DivideWith(long m, long n, List<long> primes)
        {
            while (m % n == 0)
            {
                primes.Add(n);
                m = m / n;
            }
            return m;
        }
        static List<CPair<long, long>> SoinsuBunkai(long num)
        {
            List<CPair<long, long>> primes = new List<CPair<long, long>>();
            List<long> yakusu = new List<long>();
            List<long> yakusu2 = new List<long>();
            for (int i = 2; i * i <= num; i++)
            {

                if (num % i != 0) { continue; }
                yakusu.Add(i);
                if (i * i != num) { yakusu2.Add(num / i); }
            }
            for (int i = 0; i < yakusu.Count; i++)
            {
                if (num % yakusu[i] != 0) { continue; }
                int j = 0;
                for (j = 0; num % yakusu[i] == 0; j++)
                {
                    num /= yakusu[i];
                }
                primes.Add(new CPair<long, long>(yakusu[i], j));
            }
            for (int i = yakusu2.Count - 1; i > 0; i--)
            {
                if (num % yakusu2[i] != 0) { continue; }
                int j = 0;
                for (j = 0; num % yakusu2[i] == 0; j++)
                {
                    num /= yakusu2[i];
                }
                primes.Add(new CPair<long, long>(yakusu2[i], j));
            }
            return primes;
        }
        static IEnumerable<int> Primes(int maxnum)
        {
            // var sieve = new BitArray(maxnum + 1, true);
            bool[] sieve = Enumerable.Repeat(true, maxnum + 1).ToArray();
            int squareroot = (int)Math.Sqrt(maxnum);
            for (int i = 2; i <= squareroot; i++)
            {
                if (sieve[i] == false)
                    continue;
                for (int n = i * 2; n <= maxnum; n += i)
                    sieve[n] = false;
            }
            for (int i = 2; i <= maxnum; i++)
                if (sieve[i] == true)
                    yield return i;
        }
        /*   static long[] SoinsuBunkai(long num)
           {
               if (num == 1)
                   return new long[] { 1 };
               List<long> primes = new List<long>();
               long n = DivideWith(num, 2, primes);
               n = DivideWith(n, 3, primes);
               n = DivideWith(n, 5, primes);
               long i = 0;
               while (n > 1)
               {
                   int[] ps = { 7, 11, 13, 17, 19, 23, 29, 31 };
                   foreach (var p in ps)
                   {
                       // 30m+2, 30m+3, 30m+4, 30m+5, 30m+6、30m+8... は割る必要はない。
                       n = DivideWith(n, i + p, primes);
                       if (n == 1)
                           break;
                   }
                   i += 30;
               }
               return primes.ToArray();
           }*/
        public static int h;
        public static int w;
        public static int BFS(int height, int width, char[,] map)
        {
            int[] dx = new int[] { -1, 0, 0, 1 };
            int[] dy = new int[] { 0, 1, -1, 0 };
            bool[,] visited = new bool[h, w];
            int[,] len = new int[h, w];
            var queue = new Queue<Tuple<int, int>>();
            queue.Enqueue(new Tuple<int, int>(height, width));
            int max = 0;
            visited[height, width] = true;

            while (queue.Count > 0)
            {
                var now = queue.Dequeue();
                for (int i = 0; i < 4; i++)
                {
                    int ny = now.Item1 + dy[i];
                    int nx = now.Item2 + dx[i];
                    if (ny >= h || ny < 0 || nx >= w || nx < 0)
                        continue;
                    if (visited[ny, nx] || map[ny, nx] == '#')
                        continue;
                    visited[ny, nx] = true;
                    len[ny, nx] = len[now.Item1, now.Item2] + 1;
                    max = Math.Max(max, len[ny, nx]);
                    queue.Enqueue(new Tuple<int, int>(ny, nx));
                }
            }
            return max;
        }
        static public bool Bit(long x, int dig) => ((1L << dig) & x) != 0;
        static public void Swap<T>(ref T a, ref T b)
        {
            T keep = a;
            a = b;
            b = keep;
        }
        static public void Initialize<T>(ref T[] array, T initialvalue)
        {
            array = Array.ConvertAll(array, x => initialvalue);
        }
        static int[] Manacher(string S)
        {
            int i = 0, j = 0;
            int[] R = new int[S.Length];
            while (i < S.Length)
            {
                while (i - j >= 0 && i + j < S.Length && S[i - j] == S[i + j]) ++j;
                R[i] = j;
                int k = 1;
                while (i - k >= 0 && k + R[i - k] < j) { R[i + k] = R[i - k]; ++k; }
                i += k; j -= k;
            }
            return R;
        }
        /*    static Pair[] a;
    static int SaikiZentansaku(List<int>now,int count,int last,int n,int m)
    {
        List<int> next = new List<int>();
        for (int i = 0; i < now.Count; i++)
        {
            next.Add(now[i]);
        }
        next.Add(last);
        var kore = next.ToArray();
        int ans = 0;
        if(count==n)
        {
            for (int i = 0; i < a.Length; i++)
            {
                int A = a[i].a;
                int b = a[i].b;
                int c = a[i].c;
                int d = a[i].x;
                if (kore[b] - kore[A] == c) { ans += d; }
            }
            return ans;
        }
        for (int i = last; i <= m; i++)
        {
            ans = Math.Max(ans, SaikiZentansaku(next, count + 1, i, n, m));
        }
        return ans;
    }*/
    }
    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }
        private readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private long len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) throw new EndOfStreamException();
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0)
                {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }
        public char Char()
        {
            byte b = 0;
            do b = read();
            while (b < 33 || 126 < b);
            return (char)b;
        }
        public char[] Char(long n)
        {
            char[] kore = new char[n];
            for (long i = 0; i < n; i++)
            {
                kore[i] = Char();
            }
            return kore;
        }
        public char[,] Char(long n, long m)
        {
            char[,] kore = new char[n, m];
            for (long i = 0; i < n; i++)
            {
                for (int j = 0; j < m; j++)
                {
                    kore[i, j] = Char();
                }
            }
            return kore;
        }
        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 long Long()
        {
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != '-' && (b < '0' || '9' < b));
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        public long[] Long(long n)
        {
            long[] kore = new long[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = Long();
            }
            return kore;
        }
        public Mlong[] MLong(long n)
        {
            Mlong[] kore = new Mlong[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = Long();
            }
            return kore;
        }
        public long[,] Long(long h, long w)
        {
            long[,] kore = new long[h, w];
            for (int i = 0; i < h; i++)
            {
                for (int j = 0; j < w; j++)
                {
                    kore[i, j] = Long();
                }
            }
            return kore;
        }
        public int Integer() { return (int)Long(); }
        public Mint[] MInteger(long n)
        {
            Mint[] kore = new Mint[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = Integer();
            }
            return kore;
        }
        public int[] Integer(long n)
        {
            int[] kore = new int[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = (int)Long();
            }
            return kore;
        }
        public int[,] Integer(long h, long w)
        {
            int[,] kore = new int[h, w];
            for (int i = 0; i < h; i++)
            {
                for (int j = 0; j < w; j++)
                {
                    kore[i, j] = (int)Long();
                }

            }
            return kore;
        }

        public int[] StringToInt(int n)
        {
            char[] s = Scan().ToCharArray();
            int[] kore = new int[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = s[i] - '0';
            }
            return kore;
        }
        public int[,] StringToInt(int h, int w)
        {

            int[,] kore = new int[h, w];
            for (int i = 0; i < h; i++)
            {
                char[] s = Scan().ToCharArray();
                for (int j = 0; j < w; j++)
                {
                    kore[i, j] = s[j] - '0';
                }

            }
            return kore;
        }

        public double Double() { return double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }

    public class Mlong : IComparable
    {
        public static long mod = 1000000007;
        public long num;
        public Mlong(long num) { this.num = num; }
        public override int GetHashCode()
        {
            return num.GetHashCode();
        }
        public override bool Equals(object obj)
        {
            return num.Equals(obj);
        }
        public static Mlong operator +(Mlong x, Mlong y)
        {
            return new Mlong((x.num + y.num) % mod);
        }
        public static Mlong operator -(Mlong x, Mlong y)
        {
            return new Mlong((x.num - y.num + mod) % mod);
        }
        public static Mlong operator *(Mlong x, Mlong y)
        {
            return new Mlong((long)(((1L * x.num) * y.num) % mod));
        }
        public Mlong hutujyo(Mlong x)
        {
            return new Mlong(num / x.num);
        }
        static long GetInverse(long a)
        {
            long div, p = mod, x1 = 1, y1 = 0, x2 = 0, y2 = 1;
            while (true)
            {
                if (p == 1) return x2 + mod; div = a / p; x1 -= x2 * div; y1 -= y2 * div; a %= p;
                if (a == 1) return x1 + mod; div = p / a; x2 -= x1 * div; y2 -= y1 * div; p %= a;
            }
        }
        public Mlong modjyo(Mlong x)
        {
            return num * GetInverse(x.num) % mod;
        }
        public static implicit operator Mlong(long x)
        {
            return new Mlong(x);
        }
        public static Mlong operator %(Mlong x, Mlong y)
        {
            return new Mlong(x.num % y.num);
        }
        public static explicit operator long(Mlong x)
        {
            return x.num;
        }
        public static bool operator ==(Mlong x, Mlong y)
        {
            return x.num == y.num;
        }
        public static bool operator !=(Mlong x, Mlong y)
        {
            return x.num != y.num;
        }
        public static bool operator <(Mlong x, Mlong y)
        {
            return x.num < y.num;
        }
        public static bool operator >(Mlong x, Mlong y)
        {
            return x.num > y.num;
        }
        public static bool operator <=(Mlong x, Mlong y)
        {
            return x.num <= y.num;
        }
        public static bool operator >=(Mlong x, Mlong y)
        {
            return x.num >= y.num;
        }
        public Mlong Pow(Mlong n)
        {
            Mlong pow = this;
            Mlong res = 1;
            while (n > 0)
            {
                if (n % 2 == 1) { res *= pow; }
                pow *= pow;
                n=n.hutujyo(2);
            }
            /*   if (n == 1) { return this % mod; }
               if (n % 2 == 1) { return (this * Pow(n - 1)) % mod; }
               Mint buf = Pow(n.hutujyo(2));
               return (buf * buf) % mod;*/
            return res;
        }
        public static Mlong Pow(Mlong a, Mlong n)
        {
            return a.Pow(n);
        }
        public static List<Mlong> factorialMemo = new List<Mlong>() { 1 };
        static Mlong Factorial(int x)
        {
            for (int i = factorialMemo.Count; i <= x; i++) factorialMemo.Add(factorialMemo.Last() * i);
            return factorialMemo[x];
        }
        public static Mlong Comb(Mlong n, Mlong a)
        {
            Mlong buf1 = 1;
            Mlong buf2 = 1;
            for (long i = 0; i < a.num; i++)
            {
                buf1 *= n - i;
                buf2 *= (i + 1);
            }
            return buf1.modjyo(buf2);

        }

        public static Mlong Comb(int n, int r)
        {
            return Factorial(n).modjyo(Factorial(r)).modjyo(Factorial(n - r));
        }
        public static void WriteLine(Mlong x)
        {
            Console.WriteLine(x.num);

        }

        public int CompareTo(object obj)
        {
            Mlong kore = (Mlong)obj;
            return num.CompareTo(kore.num);
        }
    }
    public class Mint : IComparable
    {
        public static int mod = 1000000007;
        public int num;
        public Mint(int Num) { num = Num; }
        public override int GetHashCode()
        {
            return num.GetHashCode();
        }
        public override bool Equals(object obj)
        {
            return num.Equals(obj);
        }
        public static Mint operator +(Mint x, Mint y)
        {
            return new Mint((x.num + y.num) % mod);
        }
        public static Mint operator -(Mint x, Mint y)
        {
            return new Mint((x.num - y.num + mod) % mod);
        }
        public static Mint operator *(Mint x, Mint y)
        {
            return new Mint((int)(((1L * x.num) * y.num) % mod));
        }
        public Mint hutujyo(Mint x)
        {
            return new Mint(num / x.num);
        }

        static int GetInverse(long a)
        {
            long div, p = mod, x1 = 1, y1 = 0, x2 = 0, y2 = 1;
            while (true)
            {
                if (p == 1) return (int)(x2 + mod); div = a / p; x1 -= x2 * div; y1 -= y2 * div; a %= p;
                if (a == 1) return (int)(x1 + mod); div = p / a; x2 -= x1 * div; y2 -= y1 * div; p %= a;
            }
        }
        public Mint modjyo(Mint x)
        {
            return num * GetInverse(x.num) % mod;
        }
        /*  public Mint modjyo(Mint x)
          {
              return num * x.Pow(mod - 2);
          }*/
        public static implicit operator Mint(int x)
        {
            return new Mint(x);
        }
        public static Mint[] Toarray(int[] x)
        {
            Mint[] buf = new Mint[x.Length];
            for (int i = 0; i < x.Length; i++)
            {
                buf[i] = x[i];
            }
            return buf;
        }
        public static explicit operator int(Mint x)
        {
            return x.num;
        }
        public static Mint operator %(Mint x, Mint y)
        {
            return new Mint(x.num % y.num);
        }
        public static bool operator ==(Mint x, Mint y)
        {
            return x.num == y.num;
        }
        public static bool operator !=(Mint x, Mint y)
        {
            return x.num != y.num;
        }
        public static bool operator <(Mint x, Mint y)
        {
            return x.num < y.num;
        }
        public static bool operator >(Mint x, Mint y)
        {
            return x.num > y.num;
        }
        public static bool operator <=(Mint x, Mint y)
        {
            return x.num <= y.num;
        }
        public static bool operator >=(Mint x, Mint y)
        {
            return x.num >= y.num;
        }
        public Mint Pow(Mint n)
        {
            Mint pow = this;
            Mint res = 1;
            while (n > 0)
            {
                if (n % 2 == 1) { res *= pow; }
                pow *= pow;
                n.hutujyo(2);
            }
            /*   if (n == 1) { return this % mod; }
               if (n % 2 == 1) { return (this * Pow(n - 1)) % mod; }
               Mint buf = Pow(n.hutujyo(2));
               return (buf * buf) % mod;*/
            return res;
        }
        public static Mint Pow(Mint a, Mint n)
        {
            return a.Pow(n);
        }
        public static List<Mint> factorialMemo = new List<Mint>() { 1 };
        static Mint Factorial(int x)
        {
            for (int i = factorialMemo.Count; i <= x; i++) factorialMemo.Add(factorialMemo.Last() * i);
            return factorialMemo[x];
        }


        public static Mint Comb(int n, int r)
        {
            return Factorial(n).modjyo(Factorial(r)).modjyo(Factorial(n - r));
        }
        public static void WriteLine(Mint x)
        {
            Console.WriteLine(x.num);

        }
        public int CompareTo(object obj)
        {
            Mint kore = (Mint)obj;
            return num.CompareTo(kore.num);
        }
    }
 
    public class Tree
    {
        public Node[] Nodes;
        public List<int[]> hen;
        public class Node
        {
            public int? parent = null;
            public List<int[]> children = new List<int[]>();
            public int parentcolor = -1;
        }
        public Tree(int kosu)
        {
            Nodes = new Node[kosu];
            for (int i = 0; i < Nodes.Length; i++)
            {
                Nodes[i] = new Node();
            }
            hen = new List<int[]>();
        }
        public void Connect(int oya, int ko, int id)
        {
            hen.Add(new int[] { oya, ko, 0 });
            Nodes[oya].children.Add(new int[] { ko, id });
            Nodes[ko].parent = oya;
        }
    }
    /// <summary>
    /// Self-Balancing Binary Search Tree
    /// (using Randamized BST)
    /// </summary>
    public class SB_BinarySearchTree<T> where T : IComparable
    {
        public class Node
        {
            public T Value;
            public Node LChild;
            public Node RChild;
            public int Count;     //size of the sub tree

            public Node(T v)
            {
                Value = v;
                Count = 1;
            }
        }

        static Random _rnd = new Random();

        public static int Count(Node t)
        {
            return t == null ? 0 : t.Count;
        }

        static Node Update(Node t)
        {
            t.Count = Count(t.LChild) + Count(t.RChild) + 1;
            return t;
        }

        public static Node Merge(Node l, Node r)
        {
            if (l == null || r == null) return l == null ? r : l;

            if ((double)Count(l) / (double)(Count(l) + Count(r)) > _rnd.NextDouble())
            {
                l.RChild = Merge(l.RChild, r);
                return Update(l);
            }
            else
            {
                r.LChild = Merge(l, r.LChild);
                return Update(r);
            }
        }

        /// <summary>
        /// split as [0, k), [k, n)
        /// </summary>
        public static Tuple<Node, Node> Split(Node t, int k)
        {
            if (t == null) return new Tuple<Node, Node>(null, null);
            if (k <= Count(t.LChild))
            {
                var s = Split(t.LChild, k);
                t.LChild = s.Item2;
                return new Tuple<Node, Node>(s.Item1, Update(t));
            }
            else
            {
                var s = Split(t.RChild, k - Count(t.LChild) - 1);
                t.RChild = s.Item1;
                return new Tuple<Node, Node>(Update(t), s.Item2);
            }
        }

        public static Node Remove(Node t, T v)
        {
            if (Find(t, v) == null) return t;
            return RemoveAt(t, LowerBound(t, v));
        }

        public static Node RemoveAt(Node t, int k)
        {
            var s = Split(t, k);
            var s2 = Split(s.Item2, 1);
            return Merge(s.Item1, s2.Item2);
        }

        public static bool Contains(Node t, T v)
        {
            return Find(t, v) != null;
        }

        public static Node Find(Node t, T v)
        {
            while (t != null)
            {
                var cmp = t.Value.CompareTo(v);
                if (cmp > 0) t = t.LChild;
                else if (cmp < 0) t = t.RChild;
                else break;
            }
            return t;
        }

        public static Node FindByIndex(Node t, int idx)
        {
            if (t == null) return null;

            var currentIdx = Count(t) - Count(t.RChild) - 1;
            while (t != null)
            {
                if (currentIdx == idx) return t;
                if (currentIdx > idx)
                {
                    t = t.LChild;
                    currentIdx -= (Count(t == null ? null : t.RChild) + 1);
                }
                else
                {
                    t = t.RChild;
                    currentIdx += (Count(t == null ? null : t.LChild) + 1);
                }
            }

            return null;
        }

        public static int UpperBound(Node t, T v)
        {
            var torg = t;
            if (t == null) return -1;

            var ret = Int32.MaxValue;
            var idx = Count(t) - Count(t.RChild) - 1;
            while (t != null)
            {
                var cmp = t.Value.CompareTo(v);

                if (cmp > 0)
                {
                    ret = Math.Min(ret, idx);
                    t = t.LChild;
                    idx -= (Count(t == null ? null : t.RChild) + 1);
                }
                else if (cmp <= 0)
                {
                    t = t.RChild;
                    idx += (Count(t == null ? null : t.LChild) + 1);
                }
            }
            return ret == Int32.MaxValue ? Count(torg) : ret;
        }

        public static int LowerBound(Node t, T v)
        {
            var torg = t;
            if (t == null) return -1;

            var idx = Count(t) - Count(t.RChild) - 1;
            var ret = Int32.MaxValue;
            while (t != null)
            {
                var cmp = t.Value.CompareTo(v);
                if (cmp >= 0)
                {
                    if (cmp == 0) ret = Math.Min(ret, idx);
                    t = t.LChild;
                    if (t == null) ret = Math.Min(ret, idx);
                    idx -= t == null ? 0 : (Count(t.RChild) + 1);
                }
                else if (cmp < 0)
                {
                    t = t.RChild;
                    idx += (Count(t == null ? null : t.LChild) + 1);
                    if (t == null) return idx;
                }
            }
            // return ret;
            return ret == Int32.MaxValue ? Count(torg) : ret;
        }

        public static Node Insert(Node t, T v)
        {
            var ub = LowerBound(t, v);
            return InsertByIdx(t, ub, v);
        }

        static Node InsertByIdx(Node t, int k, T v)
        {
            var s = Split(t, k);
            return Merge(Merge(s.Item1, new Node(v)), s.Item2);
        }

        public static IEnumerable<T> Enumerate(Node t)
        {
            var ret = new List<T>();
            Enumerate(t, ret);
            return ret;
        }

        static void Enumerate(Node t, List<T> ret)
        {
            if (t == null) return;
            Enumerate(t.LChild, ret);
            ret.Add(t.Value);
            Enumerate(t.RChild, ret);
        }
    }

    /// <summary>
    /// C-like set
    /// </summary>
    public class Set<T> where T : IComparable
    {
        protected SB_BinarySearchTree<T>.Node _root;

        public T this[int idx] { get { return ElementAt(idx); } }

        public int Count()
        {
            return SB_BinarySearchTree<T>.Count(_root);
        }

        public virtual void Insert(T v)
        {
            if (_root == null) _root = new SB_BinarySearchTree<T>.Node(v);
            else
            {
                if (SB_BinarySearchTree<T>.Find(_root, v) != null) return;
                _root = SB_BinarySearchTree<T>.Insert(_root, v);
            }
        }

        public void Clear()
        {
            _root = null;
        }

        public void Remove(T v)
        {
            _root = SB_BinarySearchTree<T>.Remove(_root, v);
        }

        public bool Contains(T v)
        {
            return SB_BinarySearchTree<T>.Contains(_root, v);
        }

        public T ElementAt(int k)
        {
            var node = SB_BinarySearchTree<T>.FindByIndex(_root, k);
            if (node == null) throw new IndexOutOfRangeException();
            return node.Value;
        }

        public int Count(T v)
        {
            return SB_BinarySearchTree<T>.UpperBound(_root, v) - SB_BinarySearchTree<T>.LowerBound(_root, v);
        }

        public int LowerBound(T v)
        {
            return SB_BinarySearchTree<T>.LowerBound(_root, v);
        }

        public int UpperBound(T v)
        {
            return SB_BinarySearchTree<T>.UpperBound(_root, v);
        }

        public Tuple<int, int> EqualRange(T v)
        {
            if (!Contains(v)) return new Tuple<int, int>(-1, -1);
            return new Tuple<int, int>(SB_BinarySearchTree<T>.LowerBound(_root, v), SB_BinarySearchTree<T>.UpperBound(_root, v) - 1);
        }

        public List<T> ToList()
        {
            return new List<T>(SB_BinarySearchTree<T>.Enumerate(_root));
        }
    }

    /// <summary>
    /// C-like multiset
    /// </summary>
    public class MultiSet<T> : Set<T> where T : IComparable
    {
        public override void Insert(T v)
        {
            if (_root == null) _root = new SB_BinarySearchTree<T>.Node(v);
            else _root = SB_BinarySearchTree<T>.Insert(_root, v);
        }
    }
    public class RollingHashSlow
    {
        public const long B = (long)1e5 + 7;
        public const long MOD = (long)1e9 + 7;
        public string S { get; set; }
        public int N { get; set; }
        public long[] Power { get; set; }
        public long[] Hash { get; set; }
        public RollingHashSlow(string s)
        {
            this.S = s;
            this.N = s.Length;
            // B^nを計算しておく
            this.Power = new long[this.N + 1];
            this.Power[0] = 1;
            for (int i = 0; i < N; i++)
            {
                this.Power[i + 1] = (this.Power[i] * B) % MOD;
            }
            // ハッシュを前計算する
            this.Hash = new long[this.N + 1];
            for (int i = 0; i < N; i++)
            {
                this.Hash[i + 1] = (this.Hash[i] * B + S[i]) % MOD;
            }
        }
        // S[l, r) のハッシュ値を求める
        public long Get(int l, int r)
        {
            var hash = this.Hash[r] - (this.Hash[l] * this.Power[r - l] % MOD);
            if (hash < 0) hash += MOD;
            return hash;
        }
    }
    class RollingHashFast
    {
        public const ulong B = (ulong)1e9 + 7;
        public string S { get; set; }
        public int N { get; set; }
        public ulong[] Power { get; set; }
        public ulong[] Hash { get; set; }
        public RollingHashFast(string s)
        {
            this.S = s;
            this.N = s.Length;
            // B^nを計算しておく
            this.Power = new ulong[this.N + 1];
            this.Power[0] = 1;
            for (int i = 0; i < N; i++)
            {
                this.Power[i + 1] = this.Power[i] * B;
            }
            // ハッシュを前計算する
            this.Hash = new ulong[this.N + 1];
            for (int i = 0; i < N; i++)
            {
                this.Hash[i + 1] = this.Hash[i] * B + S[i];
            }
        }
        // [l, r) のハッシュ値を求める
        public ulong Get(int l, int r)
        {
            var hash = this.Hash[r] - (this.Hash[l] * this.Power[r - l]);
            return hash;
        }
    }
    class Segtree<T>
    {
        int n;
        T[] tree;
        Func<T, T, T> f;
        T exnum;
        public Segtree(int m, Func<T, T, T> f, T ex)
        {
            this.f = f;
            this.exnum = ex;
            n = 1;
            while (n < m) n <<= 1;

            tree = new T[(n << 1) - 1];
            for (int i = 0; i < tree.Length; i++) tree[i] = ex;
        }
        public Segtree(int m, T ini, Func<T, T, T> f, T ex) : this(m, f, ex)
        {
            for (int i = 0; i < m; ++i) tree[i + n - 1] = ini;
            all_update();
        }
        public void assign_without_update(int j, T x)
        {
            tree[j + n - 1] = x;
        }
        public void update(int j, T x)
        {
            int i = j + n - 1;
            tree[i] = x;
            while (i > 0)
            {
                i = (i - 1) >> 1;
                tree[i] = f(tree[(i << 1) + 1], tree[(i << 1) + 2]);
            }
        }
        public void all_update()
        {
            for (int i = n - 2; i >= 0; i--)
                tree[i] = f(tree[(i << 1) + 1], tree[(i << 1) + 2]);
        }
        public T look(int i) { return tree[i + n - 1]; }

        // [s, t)
        public T run(int s, int t) { return query(s, t, 0, 0, n); }
        T query(int s, int t, int k, int l, int r)
        {
            if (r <= s || t <= l) return exnum;
            if (s <= l && r <= t) return tree[k];

            return f(query(s, t, (k << 1) + 1, l, (l + r) >> 1), query(s, t, (k + 1) << 1, (l + r) >> 1, r));
        }

    }

    class Deque<T> : IEnumerable<T>
    {
        public T this[int i]
        {
            get { return this.Buffer[(this.FirstIndex + i) % this.Capacity]; }
            set
            {
                if (i < 0) throw new ArgumentOutOfRangeException();
                this.Buffer[(this.FirstIndex + i) % this.Capacity] = value;
            }
        }
        private T[] Buffer;
        private int Capacity;
        private int FirstIndex;
        private int LastIndex
        {
            get { return (this.FirstIndex + this.Length) % this.Capacity; }
        }
        public int Length;
        public Deque(int capacity = 16)
        {
            this.Capacity = capacity;
            this.Buffer = new T[this.Capacity];
            this.FirstIndex = 0;
        }
        public void PushBack(T data)
        {
            if (this.Length == this.Capacity) this.Resize();
            this.Buffer[this.LastIndex] = data;
            this.Length++;
        }
        public void PushFront(T data)
        {
            if (this.Length == this.Capacity) this.Resize();
            var index = this.FirstIndex - 1;
            if (index < 0) index = this.Capacity - 1;
            this.Buffer[index] = data;
            this.Length++;
            this.FirstIndex = index;
        }
        public T PopBack()
        {
            if (this.Length == 0) throw new InvalidOperationException("データが空です。");
            var data = this[this.Length - 1];
            this.Length--;
            return data;
        }
        public T PopFront()
        {
            if (this.Length == 0) throw new InvalidOperationException("データが空です。");
            var data = this[0];
            this.FirstIndex++;
            this.FirstIndex %= this.Capacity;
            this.Length--;
            return data;
        }
        private void Resize()
        {
            var newArray = new T[this.Capacity * 2];
            for (int i = 0; i < this.Length; i++)
            {
                newArray[i] = this[i];
            }
            this.FirstIndex = 0;
            this.Capacity *= 2;
            this.Buffer = newArray;
        }
        public IEnumerator<T> GetEnumerator()
        {
            for (int i = 0; i < this.Length; i++)
            {
                yield return this[i];
            }
        }
        IEnumerator IEnumerable.GetEnumerator()
        {
            for (int i = 0; i < this.Length; i++)
            {
                yield return this[i];
            }
        }
    }

    class Treap<T> : IEnumerable<T> where T : IComparable
    {
        public int Length
        {
            get { return this.Root == null ? 0 : this.NodeCount(this.Root); }
        }
        public T this[int i]
        {
            get
            {
                if (this.Root.Count < i - 1) throw new IndexOutOfRangeException();
                return this.Root.Get(i);
            }
        }
        public Node Root;
        private IComparer<T> Cmp = Comparer<T>.Default;
        public Treap(IComparer<T> cmp = null)
        {
            if (cmp != null) this.Cmp = cmp;
        }
        public Node Merge(Node l, Node r)
        {
            if (l == null || r == null) return l == null ? r : l;

            if (l.Priority > r.Priority)
            {
                l.R = this.Merge(l.R, r);
                return this.Update(l);
            }
            else
            {
                r.L = this.Merge(l, r.L);
                return this.Update(r);
            }
        }
        // 左にk個、右に残り(n-k)個に分割する
        public (Node, Node) Split(int k) { return this.Split(this.Root, k); }
        public (Node, Node) Split(Node t, int k)
        {
            if (t == null) return (null, null);
            if (k <= this.NodeCount(t.L))
            {
                var s = this.Split(t.L, k);
                t.L = s.Item2;
                return (s.Item1, this.Update(t));
            }
            else
            {
                var s = this.Split(t.R, k - this.NodeCount(t.L) - 1);
                t.R = s.Item1;
                return (this.Update(t), s.Item2);
            }
        }
        public int NodeCount(Node node)
        {
            return node == null ? 0 : node.Count;
        }
        public int LowerBouond(T value) { return this.LowerBouond(this.Root, value); }
        public int LowerBouond(Node node, T value)
        {
            if (node == null) return 0;
            if (this.Cmp.Compare(value, node.Value) <= 0)
                return this.LowerBouond(node.L, value);
            else
                return this.NodeCount(node.L) + 1 + this.LowerBouond(node.R, value);
        }
        public int UpperBound(T value) { return this.UpperBound(this.Root, value); }
        public int UpperBound(Node node, T value)
        {
            if (node == null) return 0;
            if (this.Cmp.Compare(value, node.Value) >= 0)
                return this.NodeCount(node.L) + 1 + this.UpperBound(node.R, value);
            else
                return this.UpperBound(node.L, value);
        }
        public int ValueCount(T value)
        {
            return this.UpperBound(value) - this.LowerBouond(value);
        }
        public void Insert(T value)
        {
            var sub = this.Split(this.Root, this.LowerBouond(value));
            var newNode = new Node(value);
            var merged = this.Merge(sub.Item1, newNode);
            this.Root = this.Merge(merged, sub.Item2);
        }
        public void Erase(T value)
        {
            if (this.ValueCount(value) == 0) return;
            var sub = this.Split(this.Root, this.LowerBouond(value));
            this.Root = this.Merge(sub.Item1, this.Split(sub.Item2, 1).Item2);
        }
        public void InsertAt(Node node, int k)
        {
            var sub = this.Split(this.Root, k);
            var merged = this.Merge(sub.Item1, node);
            this.Root = this.Merge(merged, sub.Item2);
        }
        public void InsertAt(T value, int k)
        {
            var node = new Node(value);
            var sub = this.Split(this.Root, k);
            var merged = this.Merge(sub.Item1, node);
            this.Root = this.Merge(merged, sub.Item2);
        }
        public Node EraseAt(int k)
        {
            var sub = this.Split(this.Root, k);
            var sub2 = this.Split(sub.Item2, 1);
            var res = sub2.Item1;
            this.Root = this.Merge(sub.Item1, sub2.Item2);
            return res;
        }
        public Node Update(Node node)
        {
            node.Count = this.NodeCount(node.L) + this.NodeCount(node.R) + 1;
            return node;
        }
        public class Node
        {
            private static Random Rand = new Random();
            public T Value;
            public Node L = null;
            public Node R = null;
            public int Count;
            public double Priority;
            public Node(T value, double priority)
            {
                this.Value = value;
                this.Priority = priority;
                this.Count = 1;
            }
            public Node(T value) : this(value, Rand.NextDouble()) { }
            public T Get(int i) { return this.GetNode(i).Value; }
            public Node GetNode(int i)
            {
                var lcount = 0;
                if (this.L != null) lcount = this.L.Count;
                if (lcount == i) return this;
                else if (lcount < i) return this.R.GetNode(i - lcount - 1);
                else return this.L.GetNode(i);
            }
        }
        public IEnumerator<T> GetEnumerator()
        {
            for (int i = 0; i < this.Length; i++) yield return this.Root.Get(i);
        }
        IEnumerator IEnumerable.GetEnumerator() { return this.GetEnumerator(); }
    }
}