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;
class Program
{
  static void Main()
  {
    string S = Console.ReadLine();
    const long mod = 1000000007;
    int dig = S.Length;
    int [,] dp = new int[dig + 1, 13];
    
    // 初期化
    for (int i = 0; i < 13; i++)
    {
      string s = S.Substring(0, 1);
      if ((s == "?") && (i <= 9))
        dp[1, i] = 1;
      else if (s != "?" && int.Parse(s) == i)
        dp[1, i] = 1;
      else
        dp[1, i] = 0;
    }
    
    // dp
    for (int i = 2; i <= dig; i++)
    {
      // S の上から i 桁目の文字
      string s = S.Substring(i - 1, 1);
      
      for (int j = 0; j < 13; j++)
      {
        // 上 i 桁の %13 が j　になる場合の数 =: dp[i, j]
        
        long sum = 0;
        for (int k = 0; k < 13; k++)
        {
          int c = 0;
          if (s == "?")
          {
            int newmod = (1300 + j - k * 10) % 13;
            if (newmod >= 0 && newmod <= 9)
              c = 1;
          }
          else
          {
            int sint = int.Parse(s);
            if ((k * 10 + sint) % 13 == j)
              c = 1;
          }
          sum += (long)(dp[i-1, k] * c);
        }
        dp[i, j] = (int)(sum % mod);
      }
    }
    
    // 結果
    Console.WriteLine(dp[dig, 5]);
    
  }
  
}