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 Scala (2.11.7) (judge verdict: Accepted)

object Main {
  import java.io.{BufferedReader, InputStream, InputStreamReader}
  import java.util.StringTokenizer
  import scala.reflect.ClassTag

  def main(args: Array[String]): Unit = {
    val out = new java.io.PrintWriter(System.out)
    new Main(out, new InputReader(System.in)).solve()
    out.flush()
  }

  private val oj = System.getenv("ATCODER_DEBUG") == null
  def DEBUG(f: => Unit): Unit = {
    if (!oj){ f }
  }
  def debug(as: Array[Boolean]): Unit = if (!oj){ debug(as.map(x => if(x) "1" else "0").mkString) }
  def debug(as: Array[Int]): Unit = if (!oj){ debug(as.mkString(" ")) }
  def debug(as: Array[Long]): Unit =if (!oj){ debug(as.mkString(" ")) }
  def debugDim(m: Array[Array[Long]]): Unit = if (!oj){
    REP(m.length) { i =>
      debug(m(i))
    }
  }
  def debugDimFlip(m: Array[Array[Long]]): Unit = if (!oj){
    REP(m(0).length) { j =>
      REP(m.length) { i =>
        System.err.print(m(i)(j))
        System.err.print(" ")
      }
      System.err.println()
    }
  }
  def debug(s: => String): Unit = DEBUG {
    System.err.println(s)
  }
  def debugL(num: => Long): Unit = DEBUG {
    System.err.println(num)
  }

  class InputReader(val stream: InputStream) {
    private val reader = new BufferedReader(new InputStreamReader(stream), 32768)
    private var tokenizer: StringTokenizer = _

    def next(): String = {
      while (tokenizer == null || !tokenizer.hasMoreTokens)
        tokenizer = new StringTokenizer(reader.readLine)
      tokenizer.nextToken
    }

    def nextInt(): Int = Integer.parseInt(next())
    def nextLong(): Long = java.lang.Long.parseLong(next())
    def nextChar(): Char = next().charAt(0)

    def ni(): Int = nextInt()
    def nl(): Long = nextLong()
    def nc(): Char = nextChar()
    def ns(): String = next()
    def ns(n: Int): Array[Char] = ns().toCharArray
    def na(n: Int, offset: Int = 0): Array[Int] = map(n)(_ => ni() + offset)
    def na2(n: Int, offset: Int = 0): (Array[Int], Array[Int]) = {
      val A1, A2 = Array.ofDim[Int](n)
      REP(n) { i =>
        A1(i) = ni() + offset
        A2(i) = ni() + offset
      }
      (A1, A2)
    }
    def nm(n: Int, m: Int): Array[Array[Int]] = {
      val A = Array.ofDim[Int](n, m)
      REP(n) { i =>
        REP(m) { j =>
          A(i)(j) = ni()
        }
      }
      A
    }
    def nal(n: Int): Array[Long] = map(n)(_ => nl())
    def nm_c(n: Int, m: Int): Array[Array[Char]] = map(n) (_ => ns(m))
  }

  def REP(n: Int, offset: Int = 0)(f: Int => Unit): Unit = {
    var i = offset
    val N = n + offset
    while(i < N) { f(i); i += 1 }
  }
  def REP_r(n: Int, offset: Int = 0)(f: Int => Unit): Unit = {
    var i = n - 1 + offset
    while(i >= offset) { f(i); i -= 1 }
  }
  def TO(from: Int, to: Int)(f: Int => Unit): Unit = {
    REP(to - from + 1, from) { i =>
      f(i)
    }
  }
  def map[@specialized A: ClassTag](n: Int, offset: Int = 0)(f: Int => A): Array[A] = {
    val res = Array.ofDim[A](n)
    REP(n)(i => res(i) = f(i + offset))
    res
  }

  def sumL(as: Array[Int]): Long = {
    var s = 0L
    REP(as.length)(i => s += as(i))
    s
  }
  def cumSum(as: Array[Int]): Array[Long] = {
    val cum = Array.ofDim[Long](as.length + 1)
    REP(as.length) { i =>
      cum(i + 1) = cum(i) + as(i)
    }
    cum
  }
}

class Main(out: java.io.PrintWriter, sc: Main.InputReader) {
  import sc._
  import Main._
  import java.util.Arrays.sort

  import scala.collection.mutable
  import math.{abs, max, min}
  import mutable.ArrayBuffer

  // toIntとか+7とかするならvalにしろ
  @inline private def MOD = 1000000007

  def solve(): Unit = {
    val S = ns().reverse // 反転すると楽
    val N = S.length

    val memo = mutable.Map[(String, Boolean), Array[Long]]()
    def cnt(s: Int, t: Int, rev: Boolean): Array[Long] = {
      val str = S.substring(s, t)
      if (memo.contains((str, rev))) memo((str, rev))
      else {
        val c = Array.ofDim[Long](13)
        val lp0 = if (str(0) == '?') 10 else 1
        val lp1 = if (str.length >= 2 && str(1) == '?') 10 else 1
        val lp2 = if (str.length >= 3 && str(2) == '?') 10 else 1
        REP(lp0) { i =>
          val v0 = if (str(0) == '?') i else str(0) - '0'
          REP(lp1) { j =>
            val v1 = if (str.length >= 2 && str(1) == '?') j
              else if (str.length >= 2) str(1) - '0'
              else 0
            REP(lp2) { k =>
              val v2 = if (str.length >= 3 && str(2) == '?') k
                else if (str.length >= 3) str(2) - '0'
                else 0

              c((v0 + v1 * 10 + v2 * 100) % 13) += 1
            }
          }
        }
        if (rev) reverse(c)
        memo((str, rev)) = c
        c
      }
    }

    val dp = ArrayBuffer[Array[Long]]()
    REP((N + 3 - 1)/3) { i =>
      val c = cnt(i * 3, min(N, (i+1) * 3), i % 2 == 1)
      dp += c
    }

    debug(dp.map(_.mkString(" ")).mkString(","))
    val dp2 = Array.ofDim[Long](dp.length, 13)
    REP(13) { i =>
      dp2(0)(i) = dp(0)(i)
    }
    REP(dp2.length - 1) { i =>
      REP(13) { j =>
        REP(13) { k =>
          dp2(i + 1)((j + k)%13) += dp(i + 1)(j) * dp2(i)(k) % MOD
        }
      }
      REP(13) { j =>
        dp2(i + 1)(j) %= MOD
      }
    }

    debug(dp2.map(_.mkString(" ")).mkString(","))

    out.println(dp2.last(5))
  }

  def reverse(as: Array[Long]): Unit = {
    REP(6, 1) { i =>
      val j = 13 - i
      val t = as(i)
      as(i) = as(j)
      as(j) = t
    }
  }
}