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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.*;
import java.io.*;

public class Main {
    static int N;
    static String S;

    public static void main(String[] args) {
        S = sc.nextStr();
        N = S.length();
        int[][] dp = new int[N + 1][13];
        dp[0][0] = 1;
        for (int i = 0; i < N; i++) {
            char c = S.charAt(i);
            if (c != '?') {
                int digit = c - '0';
                for (int j = 0; j < 13; j++) {
                    int r = (10 * j + digit) % 13;
                    dp[i + 1][r] += dp[i][j];
                    dp[i + 1][r] %= MOD;
                }
            } else {
                for (int digit = 0; digit < 10; digit++) {
                    for (int j = 0; j < 13; j++) {
                        int r = (10 * j + digit) % 13;
                        dp[i + 1][r] += dp[i][j];
                        dp[i + 1][r] %= MOD;
                    }
                }
            }
        }
        long answer = dp[N][5];
        writer.println(answer);
        writer.flush();
    }

    // MOD INT SNIPPET FROM HERE
    public static final int MOD = 1_000_000_007;
    public static int[] fact;
    public static int[] invfact;

    /**
     * @param n Calculate factorials/inverse factorials for x = [0,n]
     */
    public static void preCalcFactorials(int n) {
        fact = new int[n + 1];
        invfact = new int[n + 1];
        fact[0] = 1;
        invfact[0] = 1;
        for (int i = 1; i < n + 1; i++) {
            fact[i] = modProd(i, fact[i - 1]);
            int inv = invMod(i);
            invfact[i] = modProd(inv, invfact[i - 1]);
        }
    }

    public static int properMod(int x) {
        int ans = x % MOD;
        if (ans < 0)
            ans += MOD;
        return ans;
    }

    /**
     * Calculate nCm modulo to global MOD.
     *
     * preCalcFactorials() should be executed beforehand.
     * 
     * @param n
     * @param m
     * @return combination value. 1 for n == m, 0 for n < m. -1 for n , m < 0
     */
    public static int combMod(int n, int m) {
        if (n < 0 || m < 0)
            return -1;
        if (n < m)
            return 0;
        if (n == m)
            return 1;
        int ans = modProd(fact[n], invfact[n - m]);
        ans = modProd(ans, invfact[m]);
        return ans;
    }

    /**
     * Calculate a product of 2 integers modulo to mod
     * 
     * @param x
     * @param y
     * @param mod modulo
     * @return
     */
    public static int modProd(int x, int y) {
        long ans = ((long) x) * ((long) y); // prevent overflow
        ans %= MOD;
        if (ans < 0)
            ans += MOD;
        return (int) ans;
    }

    /**
     * Calculate a power of integer modulo to mod
     * 
     * @param x
     * @param pow
     * @param mod
     * @return
     */
    public static int modPow(int x, int pow) {
        if (pow == 0)
            return 1;
        int y = modProd(x, x);
        int ans;
        if (pow % 2 == 1) {
            ans = x;
        } else {
            ans = 1;
        }
        ans = modProd(ans, modPow(y, pow / 2));
        return ans;
    }

    /**
     * Calculate a inverse of an integer modulo to mod, using Fermat's Theorem.
     * Assuming mod is a prime number.
     * 
     * @param x
     * @param mod
     * @return
     */
    public static int invMod(int x) {
        return modPow(x, MOD - 2);
    }

    /**
     * x / y modulo MOD global
     * 
     * @param x
     * @param y
     * @return
     */
    public static int modDivide(int x, int y) {
        return modProd(x, invMod(y));
    }
    // MOD INT SNIPPET END HERE

    static final FastScanner sc = new FastScanner(System.in);
    static PrintWriter writer = new PrintWriter(System.out);

    static class FastScanner {
        Reader input;

        FastScanner() {
            this(System.in);
        }

        FastScanner(InputStream stream) {
            this.input = new BufferedReader(new InputStreamReader(stream));
        }

        int nextInt() {
            return (int) nextLong();
        }

        long nextLong() {
            try {
                int sign = 1;
                int b = input.read();
                while ((b < '0' || '9' < b) && b != '-' && b != '+') {
                    b = input.read();
                }
                if (b == '-') {
                    sign = -1;
                    b = input.read();
                } else if (b == '+') {
                    b = input.read();
                }
                long ret = b - '0';
                while (true) {
                    b = input.read();
                    if (b < '0' || '9' < b)
                        return ret * sign;
                    ret *= 10;
                    ret += b - '0';
                }
            } catch (IOException e) {
                e.printStackTrace();
                return -1;
            }
        }

        double nextDouble() {
            try {
                double sign = 1;
                int b = input.read();
                while ((b < '0' || '9' < b) && b != '-' && b != '+') {
                    b = input.read();
                }
                if (b == '-') {
                    sign = -1;
                    b = input.read();
                } else if (b == '+') {
                    b = input.read();
                }
                double ret = b - '0';
                while (true) {
                    b = input.read();
                    if (b < '0' || '9' < b)
                        break;
                    ret *= 10;
                    ret += b - '0';
                }
                if (b != '.')
                    return sign * ret;
                double div = 1;
                b = input.read();
                while ('0' <= b && b <= '9') {
                    ret *= 10;
                    ret += b - '0';
                    div *= 10;
                    b = input.read();
                }
                return sign * ret / div;
            } catch (IOException e) {
                e.printStackTrace();
                return Double.NaN;
            }
        }

        char nextChar() {
            try {
                int b = input.read();
                while (Character.isWhitespace(b)) {
                    b = input.read();
                }
                return (char) b;
            } catch (IOException e) {
                e.printStackTrace();
                return 0;
            }
        }

        String nextStr() {
            try {
                StringBuilder sb = new StringBuilder();
                int b = input.read();
                while (Character.isWhitespace(b)) {
                    b = input.read();
                }
                while (b != -1 && !Character.isWhitespace(b)) {
                    sb.append((char) b);
                    b = input.read();
                }
                return sb.toString();
            } catch (IOException e) {
                e.printStackTrace();
                return "";
            }
        }
    }
    // FAST SCANNER END HERE
}