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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

def propagate(ary)
    newary = Array.new(13)
    transtable = [0, 10, 7, 4, 1, 11, 8, 5, 2, 12, 9, 6, 3]
    ary.each_with_index do |a, i|
        newary[transtable[i]] = a
    end
    newary
end
S = gets.chomp
dp = Array.new(S.length){Array.new(13, 0)}
S.each_char.with_index do |s, i|
    if s == '?'
        if i == 0
            for j in 0..9
                dp[i][j] = 1
            end
        else
            for j in 0..9
                dp[i] = dp[i].zip(propagate(dp[i-1]).rotate(-j)).map{|x, y| (x + y) % (10 ** 9 + 7)}
            end
        end
    else
        remain = s.to_i
        if i == 0
            dp[i] = Array.new(13, 0)
            dp[i][remain] = 1
        else
            dp[i] = propagate(dp[i-1]).rotate(-remain)
        end
    end
    dp[i] = dp[i].map{|num| num % (10 ** 9 + 7)}
end
puts dp[-1][5]