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 Python (3.4.3) (judge verdict: Accepted)

import numpy as np


def main():
    S = {i:s for i, s in enumerate(input()[::-1])}
    MOD = 10**9 + 7
    
    DP_table = np.zeros(13).astype(np.int64)
    
    # 初期化
    digit = S[0]
    if digit == '?':
        for i in range(10):
            DP_table[i] = 1
    else:
        DP_table[int(digit)] = 1
        
    transition_matrixes = {}
    for multiplier in range(13):
        for digit in range(10):
            transition_matrix = np.zeros((13, 13)).astype(np.int64)
            for k in range(13):
                transition_matrix[(digit * multiplier + k) % 13, k] = 1
            transition_matrixes[(multiplier, digit)] = transition_matrix

    transition_matrixes2 = {}
    for multiplier in range(13):
        transition_matrixes2[multiplier] = np.zeros((13, 13)).astype(np.int64)
        for digit in range(10):
            transition_matrixes2[multiplier] += transition_matrixes[(multiplier, digit)]
            
    # DP
    multiplier = 1
    for i in range(1, len(S)):
        multiplier = multiplier * 10 % 13
        digit = S[i]
        if digit == '?':
            transition_matrix = transition_matrixes2[multiplier]
        else:
            transition_matrix = transition_matrixes[(multiplier, int(digit))]
        
        DP_table = np.dot(transition_matrix, DP_table)
        DP_table %= MOD

    print(int(DP_table[5]))

    
if __name__ == "__main__":
    main()