Problem:
Score : 400 points

Problem Statement

Given is a string S. Each character in S is either a digit (0, ..., 9) or ?.

Among the integers obtained by replacing each occurrence of ? with a digit, how many have a remainder of 5 when divided by 13? An integer may begin with 0.

Since the answer can be enormous, print the count modulo 10^9+7.

Constraints

S is a string consisting of digits (0, ..., 9) and ?.

1 \leq |S| \leq 10^5

Input

Input is given from Standard Input in the following format:

S

Output

Print the number of integers satisfying the condition, modulo 10^9+7.

Sample Input 1

??2??5

Sample Output 1

768

For example, 482305, 002865, and 972665 satisfy the condition.

Sample Input 2

?44

Sample Output 2

1

Only 044 satisfies the condition.

Sample Input 3

7?4

Sample Output 3

0

We may not be able to produce an integer satisfying the condition.

Sample Input 4

?6?42???8??2??06243????9??3???7258??5??7???????774????4?1??17???9?5?70???76???

Sample Output 4

153716888

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System.Collections;
using System.Text;
using System.Threading.Tasks;
using static System.Math;
using static System.Console;
using System.Collections.Generic;
using System.Linq;
using System;
using System.Numerics;

class Program
{
    #region Reader
    //static Vector2Int ReadVec2Int
    //{
    //    get
    //    {
    //        string[] s = Console.ReadLine().Split(' ');
    //        return new Vector2Int(int.Parse(s[0]), int.Parse(s[1]));
    //    }
    //}
    //static Vector2Int[] ReadVec2Ints(int n)
    //{
    //    List<Vector2Int> list = new List<Vector2Int>();
    //    for (int i = 0; i < n; ++i)
    //    {
    //        var ab = ReadInts;
    //        list.Add(new Vector2Int(ab[0], ab[1]));
    //    }
    //    return list.ToArray();
    //}
    //static HashSet<Vector2Int> ReadVec2IntHash(int n)
    //{
    //    HashSet<Vector2Int> hash = new HashSet<Vector2Int>();
    //    for (int i = 0; i < n; ++i)
    //    {
    //        var ab = ReadInts;
    //        hash.Add(new Vector2Int(ab[0], ab[1]));
    //    }
    //    return hash;
    //}
    static string ReadStr => Console.ReadLine();
    static string[] ReadStrs => Console.ReadLine().Split(' ');
    static int ReadInt => Convert.ToInt32(Console.ReadLine());
    static int[] ReadInts => Console.ReadLine().Split(' ').Select(s => Convert.ToInt32(s)).ToArray();
    static long ReadLong => Convert.ToInt64(Console.ReadLine());
    static long[] ReadLongs => Console.ReadLine().Split(' ').Select(s => Convert.ToInt64(s)).ToArray();
    static long[,] ReLongMatrix(int r, int c)
    {
        var mat = new long[r, c];
        for (int i = 0; i < r; ++i)
        {
            var x = ReadLongs;
            for (int j = 0; j < c; ++j)
            {
                mat[i, j] = x[j];
            }
        }
        return mat;
    }
    static Tuple<string, string>[] ReTupleStr(int n)
    {
        List<Tuple<string, string>> list = new List<Tuple<string, string>>();
        for (int i = 0; i < n; ++i)
        {
            var ab = ReadStrs;
            list.Add(new Tuple<string, string>(ab[0], ab[1]));
        }
        return list.ToArray();
    }
    static Tuple<long, long>[] ReTupleLong(int n)
    {
        List<Tuple<long, long>> list = new List<Tuple<long, long>>();
        for (int i = 0; i < n; ++i)
        {
            var ab = ReadLongs;
            list.Add(new Tuple<long, long>(ab[0], ab[1]));
        }
        return list.ToArray();
    }
    static Tuple<int, int>[] ReadTupleInt(int n)
    {
        List<Tuple<int, int>> list = new List<Tuple<int, int>>();
        for (int i = 0; i < n; ++i)
        {
            var ab = ReadInts;
            list.Add(new Tuple<int, int>(ab[0], ab[1]));
        }
        return list.ToArray();
    }
    static Tuple<double, double>[] ReTupleDouble(int n)
    {
        List<Tuple<double, double>> list = new List<Tuple<double, double>>();
        for (int i = 0; i < n; ++i)
        {
            var ab = ReadLongs;
            list.Add(new Tuple<double, double>(ab[0], ab[1]));
        }
        return list.ToArray();
    }
    static Dictionary<int, HashSet<int>> ReAdjacencyList(int N, int M)
    {
        Dictionary<int, HashSet<int>> dict = new Dictionary<int, HashSet<int>>();
        for (int i = 0; i <= N; ++i)
        {
            dict.Add(i, new HashSet<int>());
        }
        for (int i = 0; i < M; ++i)
        {
            var ab = ReadInts;
            dict[ab[0]].Add(ab[1]);
            dict[ab[1]].Add(ab[0]);
        }
        return dict;
    }
    #endregion
    static void Swap<T>(ref T a, ref T b)
    {
        T temp = a;
        a = b;
        b = temp;
    }
    static int mod = (int)Pow(10, 9) + 7;

    static void Main()
    {
        string S = ReadStr;
        int N = S.Length;

        int[,] dp = new int[N+1, 13];

        dp[0,0] = 1;

        for (int i = 1; i <= N; i++)
        {
            int digit = N - i;
            char c = S[digit];
            if(c == '?')
            {
                for (int j = 0; j < 13; j++)
                {
                    for (int k = 0; k <= 9; k++)
                    {
                        int plus = (k * (int)PowerMod(10, i - 1, 13)) % 13;
                        dp[i, j] += dp[i - 1, (j-plus+13)%13];
                        dp[i, j] %= mod;
                    }
                }
            }
            else
            {
                int plus = ((c - '0') * (int)PowerMod(10, i - 1, 13)) % 13;
                for (int j = 0; j < 13; j++)
                {
                    dp[i, j] += dp[i - 1, (j - plus + 13) % 13];
                }
            }
        }

        WriteLine(string.Join("\n", dp[N,5]));
    }

    public static long GetInv(long a, long mod = 1000000007)
    {
        return PowerMod(a, mod - 2, mod);
    }
    public static long Combination(long a, long b, long mod = 1000000007)
    {
        if (b < 0 || b > a) return 0;
        if (b > a / 2) b = a - b;
        long numerator = Factorial(a, mod);
        long denominator = (Factorial(b, mod) * Factorial(a - b, mod)) % mod;

        return (numerator * GetInv(denominator, mod)) % mod;
    }
    public static long Factorial(long n, long mod = 1000000007)
    {
        if (n == 0) return 1;
        if (n == 1) return 1;
        long ans = n;
        for (int i = 2; i < n; i++)
        {
            ans = (ans * i) % mod;
        }

        return ans;
    }
    private static long PowerMod(long a, long n, long mod = 1000000007)
    {
        if (n == 0) return 1;
        if (n == 1) return a;

        long p = PowerMod(a, n / 2, mod);
        if (n % 2 == 1) return ((p * p) % mod * a) % mod;
        else return (p * p) % mod;
    }

}