Problem:
Score : 600 points

Problem Statement

There is an arithmetic progression with L terms: s_0, s_1, s_2, ... , s_{L-1}.

The initial term is A, and the common difference is B. That is, s_i = A + B \times i holds.

Consider the integer obtained by concatenating the terms written in base ten without leading zeros. For example, the sequence 3, 7, 11, 15, 19 would be concatenated into 37111519. What is the remainder when that integer is divided by M?

Constraints

All values in input are integers.

1 \leq L, A, B < 10^{18}

2 \leq M \leq 10^9

All terms in the arithmetic progression are less than 10^{18}.

Input

Input is given from Standard Input in the following format:

L A B M

Output

Print the remainder when the integer obtained by concatenating the terms is divided by M.

Sample Input 1

5 3 4 10007

Sample Output 1

5563

Our arithmetic progression is 3, 7, 11, 15, 19, so the answer is 37111519 mod 10007, that is, 5563.

Sample Input 2

4 8 1 1000000

Sample Output 2

891011

Sample Input 3

107 10000000000007 1000000000000007 998244353

Sample Output 3

39122908

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)


import java.math.BigInteger;

public class Main {


  private static void solve() {
    long l = nl();
    long a = nl();
    long b = nl();
    long m = ni();

    BigInteger bm = BigInteger.valueOf(m);

    long[] ten = new long[19];
    ten[0] = 1;
    for (int i = 0; i < 18; i++) {
      ten[i + 1] = ten[i] * 10;
    }


    BigInteger ret = BigInteger.ZERO;
    for (int d = 0;; d++) {
      long preFrom = (ten[d] - a + b - 1) / b;
      long preTo = (ten[d + 1] - a + b - 1) / b;
      // [from,to]

      long to = Math.min(l, preTo);
      long from = Math.max(0, preFrom);

      long len = to - from;
      BigInteger p = BigInteger.valueOf(a + b * from);
      BigInteger q = BigInteger.valueOf(b);
      BigInteger t = BigInteger.valueOf(ten[d + 1]);
      BigInteger z = BigInteger.ZERO;
      BigInteger e = BigInteger.ONE;

      BigInteger[][] mat = {
          {t, e, p},
          {z, e, q},
          {z, z, e},
      };
      
      BigInteger[] v = {ret, z, e};
      BigInteger[] x = pow(mat, v, len, bm);
      
      ret = x[0];

      if (preTo >= l)
        break;
    }

    System.out.println(ret);
  }

  public static BigInteger[] pow(BigInteger[][] A, BigInteger[] v, long e, BigInteger mod)
  {
      BigInteger[][] MUL = A;
      for(int i = 0;i < v.length;i++)v[i] = v[i].mod(mod);
      for(;e > 0;e>>>=1) {
          if((e&1)==1)v = mul(MUL, v, mod);
          MUL = p2(MUL, mod);
      }
      return v;
  }
  
  public static BigInteger[] mul(BigInteger[][] A, BigInteger[] v, BigInteger mod)
  {
      int m = A.length;
      int n = v.length;
      BigInteger[] w = new BigInteger[m];
      for(int i = 0;i < m;i++){
          BigInteger sum = BigInteger.ZERO;
          for(int k = 0;k < n;k++){
              sum = sum.add(A[i][k].multiply(v[k])).mod(mod);
          }
          w[i] = sum;
      }
      return w;
  }
  public static BigInteger[][] p2(BigInteger[][] A, BigInteger mod)
  {
      int n = A.length;
      BigInteger[][] C = new BigInteger[n][n];
      for(int i = 0;i < n;i++){
          for(int j = 0;j < n;j++){
              BigInteger sum = BigInteger.ZERO;
              for(int k = 0;k < n;k++){
                  sum = sum.add(A[i][k].multiply(A[k][j])).mod(mod);
              }
              C[i][j] = sum;
          }
      }
      return C;
  }

  public static void main(String[] args) {
    new Thread(null, new Runnable() {
      @Override
      public void run() {
        long start = System.currentTimeMillis();
        String debug = args.length > 0 ? args[0] : null;
        if (debug != null) {
          try {
            is = java.nio.file.Files.newInputStream(java.nio.file.Paths.get(debug));
          } catch (Exception e) {
            throw new RuntimeException(e);
          }
        }
        reader = new java.io.BufferedReader(new java.io.InputStreamReader(is), 32768);
        solve();
        out.flush();
        tr((System.currentTimeMillis() - start) + "ms");
      }
    }, "", 64000000).start();
  }

  private static java.io.InputStream is = System.in;
  private static java.io.PrintWriter out = new java.io.PrintWriter(System.out);
  private static java.util.StringTokenizer tokenizer = null;
  private static java.io.BufferedReader reader;

  public static String next() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
      try {
        tokenizer = new java.util.StringTokenizer(reader.readLine());
      } catch (Exception e) {
        throw new RuntimeException(e);
      }
    }
    return tokenizer.nextToken();
  }

  private static double nd() {
    return Double.parseDouble(next());
  }

  private static long nl() {
    return Long.parseLong(next());
  }

  private static int[] na(int n) {
    int[] a = new int[n];
    for (int i = 0; i < n; i++)
      a[i] = ni();
    return a;
  }

  private static char[] ns() {
    return next().toCharArray();
  }

  private static long[] nal(int n) {
    long[] a = new long[n];
    for (int i = 0; i < n; i++)
      a[i] = nl();
    return a;
  }

  private static int[][] ntable(int n, int m) {
    int[][] table = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[i][j] = ni();
      }
    }
    return table;
  }

  private static int[][] nlist(int n, int m) {
    int[][] table = new int[m][n];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[j][i] = ni();
      }
    }
    return table;
  }

  private static int ni() {
    return Integer.parseInt(next());
  }

  private static void tr(Object... o) {
    if (is != System.in)
      System.out.println(java.util.Arrays.deepToString(o));
  }
}