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 PHP7 (7.0.15) (judge verdict: Accepted)

<?php
define("MOD", 10**9+7);
$str = str_split(str());
$n = count($str);
for($j = 0; $j < 13; $j++){
  $dp[$n-1][$j] = 0;
}
if($str[$n-1] == "?"){
  for($i = 0; $i < 10; $i++){
    $dp[$n-1][$i] = 1;
  }
}else{
  $dp[$n-1][$str[$n-1]] = 1;
}
$c = 0;
$m = 1;
for($i = $n-2; $i>=0; --$i){
  $m*=10;
  $m%=13;
  for($j = 0; $j < 13; ++$j){
    $dp[$i][$j] = 0;
  }
  $dpi = &$dp[$i];
  $dpip = $dp[$i+1];
  if($str[$i] != "?"){
    $diff = $m*intval($str[$i])%13;
    for($j = 0; $j < 13; ++$j){
      $dpi[($j+$diff)%13] = $dpip[$j];
    }
  }else{
    for($k = 0; $k < 10; ++$k){
      $diff = $m*$k%13;
      for($j = 0; $j < 13; ++$j){
        $dpi[($j+$diff)%13] += $dpip[$j];
        $c+=$dpip[$j];
      }
    }
  }
  for($j = 0; $j < 13; ++$j){
    $dpi[$j] %= MOD;
  }
}
echo $dp[0][5];


function str(){
  return trim(fgets(STDIN));
}
function ints(){
  return array_map("intval", explode(" ", trim(fgets(STDIN))));
}
function int(){
  return intval(trim(fgets(STDIN)));
}
function o(...$val){
    if(count($val)==1)$val = array_shift($val);
    $trace = debug_backtrace();
    echo $trace[0]['line'].")";
    if(is_array($val)){
        if(count($val) == 0){
            echo "empty array";
        }elseif(!is_array(current($val))){
            echo "array:";
            echo implode(" ", $val)."\n";
        }else{
            echo "array:array\n";
            foreach($val as $row){
                echo implode(" ", $row)."\n";
            }
        }
    }else{
        echo $val."\n";
    }
}