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)

#if true
//#if false
#define UTAKA_DEBUG
#endif

using System;
using System.Text;
using System.Linq;
using System.Collections;
using System.Collections.Generic;
using System.IO;
using static System.Math;


namespace UtakaApp
{
    public class Program
    {
        public const string ContestName = "abc135";
        public const string ProblemName = "D";

        public const int ModLarge = 1000000007;

        public static void Main(string[] args)
        {
#if UTAKA_LOCAL
            new TestCaseChecker().TestProblems();
#else
            new Program().Solve(Console.In, new ConsoleWriter());
#endif
        }

        public void Solve(TextReader textReader, IConsoleWriter consoleWriter)
        {
            var cin = new ConsoleInput(textReader, ' ');

            string S = cin.Read;

            int Len = S.Length;

            int N = 13;

            long[][] dp = new long[Len + 1][];

            for (int i = 0; i < Len + 1; i++)
            {
                dp[i] = new long[N];
            }

            dp[0][0] = 1;

            int mul = 1;

            for (int i = 0; i < Len; i++)
            {
                int index = Len - i - 1;

                char c = S[index];

                if (c == '?')
                {
                    for (int k = 0; k < 10; k++)
                    {
                        for (int j = 0; j < N; j++)
                        {
                            dp[i + 1][(mul * k + j) % N] += dp[i][j];
                            dp[i + 1][(mul * k + j) % N] %= ModLarge;
                        }
                    }
                }
                else
                {
                    int num = c - '0';

                    for (int j = 0; j < N; j++)
                    {
                        dp[i + 1][(mul * num + j) % N] += dp[i][j];
                        dp[i + 1][(mul * num + j) % N] %= ModLarge;
                    }
                }


                for (int j = 0; j < N; j++)
                {
                    dp[i + 1][j] %= ModLarge;
                }

                mul *= 10;
                mul %= N;
            }

            consoleWriter.WriteLine(dp[Len][5].ToString());
        }
    }

    public class ConsoleInput
    {
        private readonly TextReader _stream;
        private char _separator = ' ';
        private Queue<string> inputStream;

        public ConsoleInput(TextReader stream, char separator = ' ')
        {
            this._separator = separator;
            this._stream = stream;
            inputStream = new Queue<string>();
        }

        public string Read
        {
            get
            {
                if (inputStream.Count != 0) return inputStream.Dequeue();
                string[] tmp = _stream.ReadLine().Split(_separator);
                for (int i = 0; i < tmp.Length; ++i)
                {
                    inputStream.Enqueue(tmp[i]);
                }

                return inputStream.Dequeue();
            }
        }

        public string ReadLine => _stream.ReadLine();
        public int ReadInt => int.Parse(Read);
        public long ReadLong => long.Parse(Read);
        public double ReadDouble => double.Parse(Read);

        public string[] ReadStrArray(long N)
        {
            var ret = new string[N];
            for (long i = 0; i < N; ++i) ret[i] = Read;
            return ret;
        }

        public int[] ReadIntArray(long N)
        {
            var ret = new int[N];
            for (long i = 0; i < N; ++i) ret[i] = ReadInt;
            return ret;
        }

        public long[] ReadLongArray(long N)
        {
            var ret = new long[N];
            for (long i = 0; i < N; ++i) ret[i] = ReadLong;
            return ret;
        }
    }

    public interface IConsoleWriter
    {
        void Write(string str);
        void WriteLine(string str);
    }

    public class ConsoleWriter : IConsoleWriter
    {
        public void Write(string str)
        {
            Console.Write(str);
        }

        public void WriteLine(string str)
        {
            Console.WriteLine(str);
        }
    }

#if UTAKA_LOCAL

    public class DebugConsoleWriter : IConsoleWriter
    {
        private StringBuilder mSb;

        public DebugConsoleWriter()
        {
            mSb = new StringBuilder();
        }

        public void Write(string str)
        {
            mSb.Append(str);
        }

        public void WriteLine(string str)
        {
            mSb.AppendLine(str);
        }

        public string GetAllOutput()
        {
            return mSb.ToString();
        }
    }

    public class TestCaseChecker
    {
        private string DirectoryPath => $"../../../../{Program.ContestName}/{Program.ProblemName}";

        private string GetInputFilePath(int testCaseNumber)
        {
            return $"{DirectoryPath}/in_{testCaseNumber}.txt";
        }

        private string GetOutputFilePath(int testCaseNumber)
        {
            return $"{DirectoryPath}/out_{testCaseNumber}.txt";
        }

        public bool TestProblem(int testCaseNumber)
        {
            var inputFilePath = GetInputFilePath(testCaseNumber);
            var outputFilePath = GetOutputFilePath(testCaseNumber);

            TextReader inputStream = new StreamReader(inputFilePath);

            var debugConsoleWriter = new DebugConsoleWriter();

            new Program().Solve(inputStream, debugConsoleWriter);

            var output = debugConsoleWriter.GetAllOutput();

            TextReader outputStream = new StreamReader(outputFilePath);
            var outputAnswer = outputStream.ReadToEnd();

#if UTAKA_DEBUG
            Console.Write(output);
            Console.Write(outputAnswer);
#endif

            var isCorrect = output == outputAnswer;

            return isCorrect;
        }

        public void TestProblems()
        {
            var isSuccessAll = true;

            var testCaseNumber = 1;
            while (true)
            {
                var inputFileName = GetInputFilePath(testCaseNumber);
                if (!File.Exists(inputFileName))
                {
                    break;
                }

                Console.WriteLine($"TestCase {testCaseNumber} =====================================================");

                var result = TestProblem(testCaseNumber);

                if (result)
                {
                    Console.WriteLine($"Success");
                }
                else
                {
                    isSuccessAll = false;
                    Console.WriteLine($"Failure *****");
                }

                testCaseNumber++;
            }

            if (isSuccessAll)
            {
                Console.WriteLine($"!!!!!!!!! All Success !!!!!!!!!");
            }
        }
    }

    public static class Dbg
    {
        public static void WriteLine(string str)
        {
            Console.WriteLine(str);
        }

        public static void Write(string str)
        {
            Console.Write(str);
        }
    }
#else
    public static class Dbg
    {
        public static void WriteLine(string str)
        {
        }

        public static void Write(string str)
        {
        }
    }

#endif
}