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
s = input()
effects = [1, 10, 9, 12, 3, 4]
mod = 10**9+7

amari = np.array([1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
arrays = [np.eye(13, k=i, dtype=np.int64)+np.eye(13, k=i-13, dtype=np.int64) for i in range(13)]
brrays = [sum(arrays[c*effects[i]%13] for c in range(10)) for i in range(6)]
for i, c in enumerate(reversed(s)) :
    if c != "?" :
        c = int(c)
        array = arrays[c*effects[i%6]%13]
        amari = np.dot(amari, array)
    else :
        array = brrays[i%6]
        amari = np.dot(amari, array)%mod

print(amari[5])