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: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;

public class Hello
{
    private const int MOD_N = 1000000000 + 7;

    public static void Main()
    {
        var s = Console.ReadLine();

        var dp = new int[s.Length + 1][];
        for (int i = 0; i < dp.Length; ++i)
            dp[i] = new int[13];

        dp[0][0] = 1;

        for (int i = 0; i < s.Length; ++i)
        {
            var prev = dp[i];
            var now = dp[i + 1];

            if (s[i] == '?')
            {
                for (int j = 0; j < 10; ++j)
                {
                    for (int k = 0; k < prev.Length; ++k)
                    {
                        var t = (10 * k + j) % 13;
                        now[t] += prev[k];
                        now[t] %= MOD_N;
                    }
                }
            }
            else
            {
                var c = s[i] - '0';
                for (int j = 0; j < prev.Length; ++j)
                {
                    now[(10 * j + c) % 13] = prev[j];
                }
            }
        }

        Console.WriteLine(dp[dp.Length - 1][5]);
    }

    private static int GCD(int a, int b) => a > b ? GCD_impl(a, b) : GCD_impl(b, a);
    private static int GCD_impl(int a, int b) => b == 0 ? a : GCD_impl(b, a % b);
    private static long LCM(int a, int b) => (long)a * b / GCD(a, b);

    private static int Comb(int m, int n)
    {
        var t = m - n;
        if (n > t)
            n = t;

        long numerator = 1;
        for (int i = 0; i < n; ++i)
        {
            numerator *= m - i;
            numerator %= MOD_N;
        }
        long denominator = 1;
        for (; n != 0; --n)
        {
            denominator *= n;
            denominator %= MOD_N;
        }

        var inv_denominator = ModInv((int)denominator, MOD_N);
        var ret = (int)(numerator * inv_denominator % MOD_N);

        return ret >= 0 ? ret : ret + MOD_N;
    }

    private static int ModInv(int a, int m)
    {
        int b = m, x = 1, y = 0;
        while (b != 0)
        {
            var t = a / b;
            a -= t * b;
            Swap(ref a, ref b);
            x -= t * y;
            Swap(ref x, ref y);
        }
        x %= m;
        return x >= 0 ? x : x + m;
    }

    private static void Swap<T>(ref T a, ref T b)
    {
        var t = a;
        a = b;
        b = t;
    }
}