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)

MOD = 10**9 + 7

s = gets.chomp.split('')

ss = []
s.each{ |x|
	if x == '?' then
		ss << nil
	else
		ss << x.to_i
	end
}

dp = Array.new(13, 0)
dp[0] = 1

table = Array.new(13) { Array.new(10, 0) }
(0..12).each{ |i|
	(0..9).each{ |j|
		table[i][j] = (i * 10 + j) % 13
	}
}

dp_next = []

(0 ... ss.size).each{ |i|
	dp_next = Array.new(13, 0)

	(0 ... dp.size).each { |j|
		if ss[i] == nil then
			(0..9).each{ |k|
				dp_next[table[j][k]] += dp[j]
			}
		else
			dp_next[table[j][ss[i]]] += dp[j]
		end
	}
	
	(0 ... dp_next.size).each { |j|
		if dp_next[j] > MOD then
			dp_next[j] = dp_next[j] % MOD
		end
	}
	
	dp = dp_next
}

puts dp[5] % MOD