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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <iostream>
#include <iomanip>
#include <fstream>
#include <string>
#include <array>
#include <vector>
#include <deque>
#include <list>
#include <set>
#include <map>
#include <unordered_map>
#include <unordered_set>
#include <stack>
#include <queue>
#include <bitset>
#include <tuple>
#include <cmath>
#include <complex>
#include <algorithm>
#include <utility>
#include <regex>
#include <cstdint>
#include <numeric>
#include <functional>
#include <cassert>

using namespace std;
using ll = int64_t;
using ull = uint64_t;

constexpr ll INF = 1'000'000'000'000'000'000;
namespace utils{
    template<class T, class Compare> using p_queue = priority_queue<T, vector<T>, Compare>;
    template<class T> using min_queue = p_queue<T, greater<T>>;
    template<class T> using max_queue = p_queue<T, less<T>>;

    template<class T> bool min_update(T& X, const T& A){ if(X > A) {X = A; return true;} return false; }
    template<class T> bool max_update(T& X, const T& A){ if(X < A) {X = A; return true;} return false; }
    using V_Set = unordered_set<int>;
    using E_Set = unordered_map<int, V_Set>;

#define ALL(x) begin(x), end(x)
#define rALL(x) rbegin(x), rend(x)
}
using namespace utils;

void sub_main(istream &is) {
    string S;
    is >> S;
    if(!is) return;
    reverse(ALL(S));

    constexpr ll MOD = 1'000'000'007;
    ll dig = 1;
    vector<array<ll, 13>> dp(S.size()+1);
    dp[0][0] = 1;

    for (int i = 0; i < S.size(); ++i) {
        for (int j = 0; j < 13; ++j) {
            if(S[i] == '?') {
                for (int k = 0; k < 10; ++k) {
                    ll f = k * dig;
                    f %= 13;
                    ll g = j - f + 13;
                    g %= 13;
                    dp[i + 1][j] += dp[i][g];
                } // end k
            }
            else{
                ll k = S[i] - '0';
                ll f = k * dig;
                f %= 13;
                ll g = j - f + 13;
                g %= 13;
                dp[i+1][j] += dp[i][g];
            }
            dp[i+1][j] %= MOD;
        } // end j
        dig *= 10;
        dig %= 13;
    } // end i

    cout << dp.back().at(5) << endl;

}

int main(int argc, char *argv[]) {
    cin.tie(nullptr);
    ios::sync_with_stdio(false);
    cout << setprecision(16) << scientific;
#ifdef DEBUG
    string test_cases = "test_D.txt";
    cerr << "DEBUG MODE" <<  endl;
    cerr << test_cases << " -->" << endl;
    auto fs = fstream(test_cases, fstream::in);
    int lp = 0;
    while(fs) {
        lp++;
        cout << lp <<  "#------\n";
        sub_main(fs);
    }
    cout << "------#" << endl;
    if(lp <= 1) sub_main(cin);
#else
    sub_main(cin);
#endif
    return 0;
}