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 PyPy3 (2.4.0) (judge verdict: Accepted)

import itertools
S = input()[::-1]
mod = 10**9 + 7
A = [0] * 13
N = 0
for i, v in enumerate(S):
    p = pow(10, i, 13)
    if v == '?':
        A[p] += 1
    else:
        N += int(v) * p
        N %= 13
dp = [0] * 13
dp[N] = 1
for i, v in enumerate(A):
    for _ in range(v):
        dp_old = dp[:]
        for j, k in itertools.product(range(1, 10), range(13)):
            x = (j * i + k) % 13
            dp[x] += dp_old[k]
            dp[x] %= mod
print(dp[5])