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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
#define mod 1000000007

long long int dp[100005][14]= {0}, arr[100005] ={0};

long long int parade(int i, int n, int sum){
    if(i==n) return (sum==5);
    
    if(dp[i][sum]!=-1) return dp[i][sum];
    
    long long int ans = 0;
    if(arr[i]!=-1){
        ans = ((ans%mod) + (parade(i+1, n, (sum*10+arr[i])%13)%mod))%mod;
        return dp[i][sum] = ans%mod;
    }

    for(int j = 0; j < 10; j++){
        ans = ((ans%mod) + (parade(i+1, n, (sum*10+j)%13)%mod))%mod;
    }
    return dp[i][sum] = ans%mod;
}

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    
    string s;
    cin >> s;
    
    for(int i = 0; i < s.size(); i++){
        if(s[i]=='?') arr[i] = -1;
        else arr[i] = s[i] -'0';
    }
    memset(dp, -1, sizeof(dp));
    cout << parade(0, s.size(), 0)%mod;
    return 0;
}