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)

# refer to https://www.planeta.tokyo/abc-135
# refer to https://atcoder.jp/contests/abc135/submissions/6599605
mod = 10 ** 9 + 7
def an(n,d=0) Array.new(n,d)end
def an2(n,m,d=0) Array.new(n){an(m,d)}end
def an3(n,m,l,d=0) Array.new(n){an2(m,l,d)}end

s = gets.chomp.chars

dp = Array.new(13){0}
dp1 = Array.new(13){0}
dp[0] = 1

m = 1

s.reverse_each.with_index do |c,i|
  i.even? ? (dp1 = Array.new(13){0}) : (dp = Array.new(13){0})
  if c == '?'
    10.times do |k|
      13.times do |j|
        jn = (m * k + j) % 13
        if i.even?
          dp1[jn] = (dp1[jn] + dp[j]) % mod
        else
          dp[jn] = (dp[jn] + dp1[j]) % mod
        end
      end
    end
  else
    13.times do |j|
      jn = (m * c.to_i + j) % 13
      if i.even?
        dp1[jn] = (dp1[jn] + dp[j]) % mod
      else
        dp[jn] = (dp[jn] + dp1[j]) % mod
      end
    end
  end
  m = m * 10 % 13
end

puts s.size.even? ? dp[5] : dp1[5]