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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;

class D
{
	static void Main()
	{
		var r = Enumerable.Range(0, 13);
		var d = Enumerable.Range(0, 10).Select(i =>
		{
			var m = new int[13, 13];
			foreach (var j in r) m[(10 * j + i) % 13, j] = 1;
			return new { i, m };
		})
		.ToDictionary(_ => _.i.ToString()[0], _ => _.m);

		var m_ = new int[13, 13];
		foreach (var p in r.SelectMany(i => r.Select(j => new { i, j }))) m_[p.i, p.j] = d.Values.Sum(m => m[p.i, p.j]);
		d['?'] = m_;

		var x = new long[13];
		x[0] = 1;
		x = Console.ReadLine().Aggregate(x, (v, c) => r.Select(i => r.Sum(j => d[c][i, j] * v[j]) % 1000000007).ToArray());
		Console.WriteLine(x[5]);
	}
}