import java.util.*;
import java.util.zip.*;
import java.util.List;
import java.util.regex.*;
import java.util.concurrent.*;
import java.util.concurrent.atomic.*;
import java.util.concurrent.locks.*;
import java.util.function.*;
import javax.swing.*;
import javax.swing.event.*;
import javax.swing.text.*;
import javax.swing.table.*;
import java.io.*;
import java.net.*;
import java.lang.reflect.*;
import java.lang.ref.*;
import java.lang.management.*;
import java.security.*;
import java.security.spec.*;
import java.awt.*;
import java.awt.event.*;
import java.awt.image.*;
import java.awt.geom.*;
import javax.imageio.*;
import java.math.*;
import java.time.Duration;
import java.lang.invoke.VarHandle;
import java.lang.invoke.MethodHandles;
class main {

static class HashedByteArray implements Comparable<HashedByteArray> {
  byte[] data;
  int hashCode;
  
  HashedByteArray() {}
  HashedByteArray(byte[] data) {
  this.data = data;}
  
  public int hashCode() {
    if (hashCode == 0)
      hashCode = oneIfZero(arrayHashCode(data));
    return hashCode;
  }
  
  public boolean equals(Object o) {
    if (o instanceof HashedByteArray)
      return this == ((HashedByteArray) o) || hashCode() == ((HashedByteArray) o).hashCode() && byteArraysEqual(data, ((HashedByteArray) o).data);
    return false;
  }
  
  // we interpret the bytes as ubytes
  public int compareTo(HashedByteArray a) {
    return Arrays.compareUnsigned(data, a.data);
  }
  
  public String toString() {
    return bytesToHex(data);
  }
}
static int oneIfZero(int i) {
  return i == 0 ? 1 : i;
}


static int arrayHashCode(byte[] a) {
  return Arrays.hashCode(a);
}


static int hashCode(Object a) {
  return a == null ? 0 : a.hashCode();
}

static int hashCode(long l) {
  return Long.hashCode(l);
}

static int hashCode(double d) {
  return Double.hashCode(d);
}


static boolean byteArraysEqual(byte[] a, byte[] b) {
  return Arrays.equals(a, b);
}


  public static String bytesToHex(byte[] bytes) {
    return bytesToHex(bytes, 0, bytes.length);
  }

  public static String bytesToHex(byte[] bytes, int ofs, int len) {
    StringBuilder stringBuilder = new StringBuilder(len*2);
    for (int i = 0; i < len; i++) {
      String s = "0" + Integer.toHexString(bytes[ofs+i]);
      stringBuilder.append(s.substring(s.length()-2, s.length()));
    }
    return stringBuilder.toString();
  }




}