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.Scanner;

public class Main {

    public static void main(String[] args) {

        Scanner scanner = new Scanner(System.in);

        String pattern = scanner.next();


        int[][] memo = new int[pattern.length()][13];
        int[] remTable = new int[pattern.length()];

        remTable[remTable.length - 1] = 1;

        for (int i = remTable.length - 2; i >= 0; i--) {
            remTable[i] = (remTable[i + 1] * 10) % 13;
        }

        System.out.println(run(pattern, 0, remTable, 5, memo));
    }

    private static int modValue = (int) (Math.pow(10, 9) + 7);

    private static int run(String pattern, int i, int[] remTable, int rem, int[][] memo) {
        if (i == pattern.length()) {
            return rem == 0 ? 1 : 0;
        }

        if (memo[i][rem] != 0) {
            return Math.max(0, memo[i][rem]);
        }

        int d = Character.getNumericValue(pattern.charAt(i));

        int sum = 0;

        if (d == -1) { // '?' case
            for (int j = 0; j <= 9; j++) {
                int actualRem = (j * remTable[i]) % 13;
                int newRem = rem - actualRem;
                if (newRem < 0) newRem += 13;
                sum += run(pattern, i + 1, remTable, newRem, memo);
                sum = sum % modValue;
            }
        } else {
            int actualRem = (d * remTable[i]) % 13;
            int newRem = rem - actualRem;
            if (newRem < 0) newRem += 13;
            sum += run(pattern, i + 1, remTable, newRem, memo);
            sum = sum % modValue;
        }

        if (sum == 0) {
            memo[i][rem] = -1;
        } else {
            memo[i][rem] = sum;
        }

        return sum;
    }
}