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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using static System.Console;
using static System.Math;

class MainClass
{
    static void Main()
    {
        string S = ReadLine();
        int N = S.Length;
        var dp = new long[N+1, 13];
        dp[0, 0] = 1;
        for (int i = 0;i<N;i++)
        {
            var s = S[N-1-i] - '0';
            if (0 <= s && s < 10)
            {
                Modular next = s* Modular.Pow(10, i);
                for (int j = 0; j < 13; j++)
                {
                    
                    dp[i + 1, (int)(next + j)] += dp[i, j];
                }
            }
            else
            {
                for (int j = 0; j < 10; j++)
                {
                    Modular next = j* Modular.Pow(10, i);
                    for (int k = 0; k < 13; k++)
                    {

                        dp[i + 1, (int)(next + k)] += dp[i, k];
                        dp[i + 1, (int)(next + k)] %= 1_000_000_007;
                    }
                }
            }
        }
        WriteLine(dp[N,5]);
    }
}
class Modular
{
    private const int M = 13;
    private long value;
    public Modular(long value = 0) { this.value = value; }
    public static implicit operator Modular(long a)
    {
        var m = a % M;
        return new Modular((m < 0) ? m + M : m);
    }
    public static Modular operator +(Modular a, Modular b)
    => a.value + b.value;
    public static Modular operator -(Modular a, Modular b)
    => a.value - b.value;
    public static Modular operator *(Modular a, Modular b)
    => a.value * b.value;
    public static Modular Pow(Modular a, int n)
    {
        switch (n)
        {
            case 0:
                return 1;
            case 1:
                return a;
            default:
                var p = Pow(a, n / 2);
                return p * p * Pow(a, n % 2);
        }
    }
    public static Modular operator /(Modular a, Modular b)
    {
        return a * Pow(b, M - 2);
    }
    private static readonly List<int> facs = new List<int> { 1 };
    public static Modular Fac(int n)   //階乗
    {
        for (int i = facs.Count; i <= n; ++i)
        {
            facs.Add((int)(Math.BigMul(facs.Last(), i) % M));
        }
        return facs[n];
    }
    public static Modular Fac(int r, int n)
    {
        int temp = 1;
        for (int i = r; i <= n; i++)
        {
            temp = (int)(Math.BigMul(temp, i) % M);
        }
        return temp;
    }
    public static Modular Ncr(int n, int r) //nCr
    {
        return (n < r) ? 0
             : (n == r) ? 1
                        : (n < 1000000) ? Fac(n) / (Fac(r) * Fac(n - r))
                            : Fac(n - r + 1, n) / Fac(r);
    }
    public static explicit operator int(Modular a)
    {
        return (int)a.value;
    }
}