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>
#define rep(i,n) for (int i = 0; i < n; i++)
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
typedef pair<int, int> pi;
typedef pair<pi, pi> pp;
typedef pair<ll, ll> pl;
const double EPS = 1e-9;
const ll MOD = 1000000007;
const int inf = 1 << 30;
const ll linf = 1LL << 60;

const int M = 13;

string s;
ll t = 1, ans;
ll dp[2][13];
int ind[13][13];
int ho = 0, ge = 1;

int main() {
    cin.tie(0);
    ios::sync_with_stdio(false);

    rep(i,M) {
        rep(j,M) ind[i][j] = i*j%M;
    }

    cin >> s;
    dp[ho][0] = 1;
    for (int i = (int)s.size()-1; i >= 0; i--) {
        if (s[i] != '?') {
            int x = t*(s[i]-'0')%M;
            rep(j,M) {
                if (x == M) x -= M;
                dp[ge][x++] = dp[ho][j];
            }
        } else {
            ll sum = 0;
            rep(j,9) sum += dp[ho][ind[t][j]];
            for (int j = 9; j < M; j++) {
                sum += dp[ho][ind[t][j]];
                dp[ge][ind[t][j]] = sum;
                sum = sum+100*MOD-dp[ho][ind[t][j-9]];
            }
            for (int j = 0; j < 9; j++) {
                sum += dp[ho][ind[t][j]];
                dp[ge][ind[t][j]] = sum;
                sum = sum+100*MOD-dp[ho][ind[t][j+4]];
            }
        }
        if (i%3) rep(j,M) dp[ge][j] %= MOD;
        t = t * 10 % M;
        swap(ho,ge);
    }
    cout << dp[ho][5]%MOD << endl;
}