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 <A> List<A> takeFirst(List<A> l, int n) {
  return l(l) <= n ? l : newSubListOrSame(l, 0, n);
}

static <A> List<A> takeFirst(int n, List<A> l) {
  return takeFirst(l, n);
}

static String takeFirst(int n, String s) { return substring(s, 0, n); }
static String takeFirst(String s, int n) { return substring(s, 0, n); }

static CharSequence takeFirst(int n, CharSequence s) { return subCharSequence(s, 0, n); }

static <A> List<A> takeFirst(int n, Iterator<A> it) {
  if (it == null) return null;
  List l = new ArrayList();
  for (int _repeat_0 = 0; _repeat_0 < n; _repeat_0++)  { if (it.hasNext()) l.add(it.next()); else break; }
  return l;
}

static <A> List<A> takeFirst(int n, Iterable<A> i) {
  if (i == null) return null;
  return i == null ? null : takeFirst(n, i.iterator());
}

static <A> List<A> takeFirst(int n, IterableIterator<A> i) {
  return takeFirst(n, (Iterator<A>) i);
}

static int[] takeFirst(int n, int[] a) { return takeFirstOfIntArray(n, a); }

static short[] takeFirst(int n, short[] a) { return takeFirstOfShortArray(n, a); }

static byte[] takeFirst(int n, byte[] a) { return takeFirstOfByteArray(n, a); }
static byte[] takeFirst(byte[] a, int n) { return takeFirstOfByteArray(n, a); }

static double[] takeFirst(int n, double[] a) { return takeFirstOfDoubleArray(n, a); }
static double[] takeFirst(double[] a, int n) { return takeFirstOfDoubleArray(n, a); }

static <A, B> Map<A, B> takeFirst(int n, Map<A, B> map) {
  return takeFirstFromMap(n, map);
}
static int l(Object[] a) { return a == null ? 0 : a.length; }
static int l(boolean[] a) { return a == null ? 0 : a.length; }
static int l(byte[] a) { return a == null ? 0 : a.length; }
static int l(short[] a) { return a == null ? 0 : a.length; }
static int l(long[] a) { return a == null ? 0 : a.length; }
static int l(int[] a) { return a == null ? 0 : a.length; }
static int l(float[] a) { return a == null ? 0 : a.length; }
static int l(double[] a) { return a == null ? 0 : a.length; }
static int l(char[] a) { return a == null ? 0 : a.length; }
static int l(Collection c) { return c == null ? 0 : c.size(); }

static int l(Iterator i) { return iteratorCount_int_close(i); } // consumes the iterator && closes it if possible

static int l(Map m) { return m == null ? 0 : m.size(); }
static int l(CharSequence s) { return s == null ? 0 : s.length(); }
static long l(File f) { return f == null ? 0 : f.length(); }



























static <A> List<A> newSubListOrSame(List<A> l, int startIndex) {
  return newSubListOrSame(l, startIndex, l(l));
}

static <A> List<A> newSubListOrSame(List<A> l, int startIndex, int endIndex) {
  if (l == null) return null;
  int n = l(l);
  startIndex = max(0, startIndex);
  endIndex = min(n, endIndex);
  if (startIndex >= endIndex) return ll();
  if (startIndex == 0 && endIndex == n) return l;
  return cloneList(l.subList(startIndex, endIndex));
}




static String substring(String s, int x) {
  return substring(s, x, strL(s));
}

static String substring(String s, int x, int y) {
  if (s == null) return null;
  if (x < 0) x = 0;
  int n = s.length();
  if (y < x) y = x;
  if (y > n) y = n;
  if (x >= y) return "";
  return s.substring(x, y);
}



// convenience method for quickly dropping a prefix
static String substring(String s, CharSequence l) {
  return substring(s, lCharSequence(l));
}


static CharSequence subCharSequence(CharSequence s, int x) {
  return subCharSequence(s, x, s == null ? 0 : s.length());
}

static CharSequence subCharSequence(CharSequence s, int x, int y) {
  if (s == null) return null;
  if (x < 0) x = 0;
  if (x >= s.length()) return "";
  if (y < x) y = x;
  if (y > s.length()) y = s.length();
  return s.subSequence(x, y);
}


static int[] takeFirstOfIntArray(int[] b, int n) {
  return subIntArray(b, 0, n);
}

static int[] takeFirstOfIntArray(int n, int[] b) {
  return takeFirstOfIntArray(b, n);
}


static short[] takeFirstOfShortArray(short[] b, int n) {
  return subShortArray(b, 0, n);
}

static short[] takeFirstOfShortArray(int n, short[] b) {
  return takeFirstOfShortArray(b, n);
}


static byte[] takeFirstOfByteArray(byte[] b, int n) {
  return subByteArray(b, 0, n);
}

static byte[] takeFirstOfByteArray(int n, byte[] b) {
  return takeFirstOfByteArray(b, n);
}


static double[] takeFirstOfDoubleArray(double[] b, int n) {
  return subDoubleArray(b, 0, n);
}

static double[] takeFirstOfDoubleArray(int n, double[] b) {
  return takeFirstOfDoubleArray(b, n);
}


static <A, B> Map<A, B> takeFirstFromMap(int n, Map<A, B> map) {
  if (map == null) return null;
  Map<A, B> map2 = similarEmptyMap(map);
  if (n > 0)
    for (Map.Entry<A, B> e : map.entrySet())
      if (n-- <= 0) break; else map2.put(e.getKey(), e.getValue());
  return map2;
}




static <A> int iteratorCount_int_close(Iterator<A> i) { try {
  int n = 0;
  if (i != null) while (i.hasNext()) { i.next(); ++n; }
  if (i instanceof AutoCloseable) ((AutoCloseable) i).close();
  return n;
} catch (Exception __e) { throw rethrow(__e); } }


static int max(int a, int b) { return Math.max(a, b); }
static int max(int a, int b, int c) { return max(max(a, b), c); }
static long max(int a, long b) { return Math.max((long) a, b); }
static long max(long a, long b) { return Math.max(a, b); }
static double max(int a, double b) { return Math.max((double) a, b); }
static float max(float a, float b) { return Math.max(a, b); }
static double max(double a, double b) { return Math.max(a, b); }

static <A extends Comparable<A>> A max (Iterable<A> l) {
  A max = null;
  var it = iterator(l);
  if (it.hasNext()) {
    max = it.next();
    while (it.hasNext()) {
      A a = it.next();
      if (cmp(a, max) > 0)
        max = a;
    }
  }
  return max;
}

/*Nah.
static int max(Collection<Integer> c) {
  int x = Integer.MIN_VALUE;
  for (int i : c) x = max(x, i);
  ret x;
}*/

static double max(double[] c) {
  if (c.length == 0) return Double.MIN_VALUE;
  double x = c[0];
  for (int i = 1; i < c.length; i++) x = Math.max(x, c[i]);
  return x;
}

static float max(float[] c) {
  if (c.length == 0) return Float.MAX_VALUE;
  float x = c[0];
  for (int i = 1; i < c.length; i++) x = Math.max(x, c[i]);
  return x;
}

static byte max(byte[] c) {
  byte x = -128;
  for (byte d : c) if (d > x) x = d;
  return x;
}

static short max(short[] c) {
  short x = -0x8000;
  for (short d : c) if (d > x) x = d;
  return x;
}

static int max(int[] c) {
  int x = Integer.MIN_VALUE;
  for (int d : c) if (d > x) x = d;
  return x;
}

static <A extends Comparable<A>> A max(A a, A b) {
  return cmp(a, b) >= 0 ? a : b;
}


static int min(int a, int b) {
  return Math.min(a, b);
}

static long min(long a, long b) {
  return Math.min(a, b);
}

static float min(float a, float b) { return Math.min(a, b); }
static float min(float a, float b, float c) { return min(min(a, b), c); }

static double min(double a, double b) {
  return Math.min(a, b);
}

static double min(double[] c) {
  double x = Double.MAX_VALUE;
  for (double d : c) x = Math.min(x, d);
  return x;
}

static float min(float[] c) {
  float x = Float.MAX_VALUE;
  for (float d : c) x = Math.min(x, d);
  return x;
}

static byte min(byte[] c) {
  byte x = 127;
  for (byte d : c) if (d < x) x = d;
  return x;
}

static short min(short[] c) {
  short x = 0x7FFF;
  for (short d : c) if (d < x) x = d;
  return x;
}

static int min(int[] c) {
  int x = Integer.MAX_VALUE;
  for (int d : c) if (d < x) x = d;
  return x;
}

static <A extends Comparable<A>> A min(A a, A b) {
  return cmp(a, b) <= 0 ? a : b;
}


static <A> List<A> ll(A... a) {
  ArrayList l = new ArrayList(a.length);
  if (a != null) for (A x : a) l.add(x);
  return l;
}


static <A> ArrayList<A> cloneList(Iterable<A> l) {
  return l instanceof Collection ? cloneList((Collection) l) : asList(l);
}

static <A> ArrayList<A> cloneList(Collection<A> l) {
  if (l == null) return new ArrayList();
  synchronized(collectionMutex(l)) {
    return new ArrayList<A>(l);
  }
}


static int strL(String s) {
  return s == null ? 0 : s.length();
}


static int lCharSequence(CharSequence s) {
  return s == null ? 0 : s.length();
}


static int[] subIntArray(int[] b, int start) {
  return subIntArray(b, start, l(b));
}
  
static int[] subIntArray(int[] b, int start, int end) {
  start = max(start, 0); end = min(end, l(b));
  if (start == 0 && end == l(b)) return b;
  if (start >= end) return new int[0];
  int[] x = new int[end-start];
  System.arraycopy(b, start, x, 0, end-start);
  return x;
}




static short[] subShortArray(short[] b, int start, int end) {
  start = max(start, 0); end = min(end, l(b));
  if (start == 0 && end == l(b)) return b;
  if (start >= end) return new short[0];
  short[] x = new short[end-start];
  System.arraycopy(b, start, x, 0, end-start);
  return x;
}


static byte[] subByteArray(byte[] b, int start) {
  return subByteArray(b, start, l(b));
}
  
static byte[] subByteArray(byte[] b, int start, int end) {
  start = max(start, 0); end = min(end, l(b));
  if (start == 0 && end == l(b)) return b;
  if (start >= end) return new byte[0];
  byte[] x = new byte[end-start];
  System.arraycopy(b, start, x, 0, end-start);
  return x;
}




static double[] subDoubleArray(double[] b, int start) { return subDoubleArray(b, start, l(b)); }
static double[] subDoubleArray(double[] b, int start, int end) {
  start = max(start, 0); end = min(end, l(b));
  if (start == 0 && end == l(b)) return b;
  if (start >= end) return new double[0];
  double[] x = new double[end-start];
  System.arraycopy(b, start, x, 0, end-start);
  return x;
}


static Map similarEmptyMap(Map m) {
  if (m instanceof TreeMap) return new TreeMap(((TreeMap) m).comparator());
  if (m instanceof LinkedHashMap) return new LinkedHashMap();
  
  // default to a hash map
  return new HashMap();
}

static Map similarEmptyMap(Iterable m) {
  if (m instanceof TreeSet) return new TreeMap(((TreeSet) m).comparator());
  if (m instanceof LinkedHashSet) return new LinkedHashMap();
  
  return new HashMap();
}




static RuntimeException rethrow(Throwable t) {
  
  if (t instanceof Error)
    _handleError((Error) t);
  
  throw t instanceof RuntimeException ? (RuntimeException) t : new RuntimeException(t);
}

static RuntimeException rethrow(String msg, Throwable t) {
  throw new RuntimeException(msg, t);
}


static <A> Iterator<A> iterator(Iterable<A> c) {
  return c == null ? emptyIterator() : c.iterator();
}


static int cmp(Number a, Number b) {
  return a == null ? b == null ? 0 : -1 : cmp(a.doubleValue(), b.doubleValue());
}

static int cmp(double a, double b) {
  return a < b ? -1 : a == b ? 0 : 1;
}

static int cmp(int a, int b) {
  return a < b ? -1 : a == b ? 0 : 1;
}

static int cmp(long a, long b) {
  return a < b ? -1 : a == b ? 0 : 1;
}

static int cmp(Object a, Object b) {
  if (a == null) return b == null ? 0 : -1;
  if (b == null) return 1;
  return ((Comparable) a).compareTo(b);
}


// unclear semantics as to whether return null on null

static <A> ArrayList<A> asList(A[] a) {
  return a == null ? new ArrayList<A>() : new ArrayList<A>(Arrays.asList(a));
}

static ArrayList<Character> asList(char[] a) {
  if (a == null) return null;
  ArrayList<Character> l = emptyList(a.length);
  for (var i : a) l.add(i);
  return l;
}

static ArrayList<Byte> asList(byte[] a) {
  if (a == null) return null;
  ArrayList<Byte> l = emptyList(a.length);
  for (var i : a) l.add(i);
  return l;
}

static ArrayList<Integer> asList(int[] a) {
  if (a == null) return null;
  ArrayList<Integer> l = emptyList(a.length);
  for (int i : a) l.add(i);
  return l;
}

static ArrayList<Long> asList(long[] a) {
  if (a == null) return null;
  ArrayList<Long> l = emptyList(a.length);
  for (long i : a) l.add(i);
  return l;
}

static ArrayList<Float> asList(float[] a) {
  if (a == null) return null;
  ArrayList<Float> l = emptyList(a.length);
  for (float i : a) l.add(i);
  return l;
}

static ArrayList<Double> asList(double[] a) {
  if (a == null) return null;
  ArrayList<Double> l = emptyList(a.length);
  for (double i : a) l.add(i);
  return l;
}

static ArrayList<Short> asList(short[] a) {
  if (a == null) return null;
  ArrayList<Short> l = emptyList(a.length);
  for (short i : a) l.add(i);
  return l;
}

static <A> ArrayList<A> asList(Iterator<A> it) {
  ArrayList l = new ArrayList();
  if (it != null)
    while (it.hasNext())
      l.add(it.next());
  return l;  
}

// disambiguation
static <A> ArrayList<A> asList(IterableIterator<A> s) {
  return asList((Iterator) s);
}

static <A> ArrayList<A> asList(Iterable<A> s) {
  if (s instanceof ArrayList) return (ArrayList) s;
  ArrayList l = new ArrayList();
  if (s != null)
    for (A a : s)
      l.add(a);
  return l;
}



static <A> ArrayList<A> asList(Enumeration<A> e) {
  ArrayList l = new ArrayList();
  if (e != null)
    while (e.hasMoreElements())
      l.add(e.nextElement());
  return l;
}






// TODO: JDK 17!! ?? No! Yes? Yes!!

static Object collectionMutex(List l) {
  return l;
}

static Object collectionMutex(Object o) {
  

  if (o instanceof List) return o;
  
  // TODO: actually use our own maps so we can get the mutex properly
  
  String c = className(o);
  
    
  
    
  
  return o;
}




static void _handleError(Error e) {
  //call(javax(), '_handleError, e);
}


static Iterator emptyIterator() {
  return Collections.emptyIterator();
}


static ArrayList emptyList() {
  return new ArrayList();
  //ret Collections.emptyList();
}

static ArrayList emptyList(int capacity) {
  return new ArrayList(max(0, capacity));
}

// Try to match capacity
static ArrayList emptyList(Iterable l) {
  return l instanceof Collection ? emptyList(((Collection) l).size()) : emptyList();
}

static ArrayList emptyList(Object[] l) {
  return emptyList(l(l));
}

// get correct type at once
static <A> ArrayList<A> emptyList(Class<A> c) {
  return new ArrayList();
}


static String className(Object o) {
  return getClassName(o);
}




static String getClassName(Object o) {
  return o == null ? "null" : o instanceof Class ? ((Class) o).getName() : o.getClass().getName();
}




// you still need to implement hasNext() and next()
static abstract class IterableIterator<A> implements Iterator<A>, Iterable<A> {
  public Iterator<A> iterator() {
    return this;
  }
  
  public void remove() {
    unsupportedOperation();
  }
}


static UnsupportedOperationException unsupportedOperation() {
  throw new UnsupportedOperationException();
}



}