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 static x30_pkg.x30_util.DynamicObject;
import java.nio.file.Path;
import java.text.*;
import java.text.NumberFormat;
import java.awt.geom.*;
import java.nio.charset.Charset;
import java.util.TimeZone;
import java.text.SimpleDateFormat;
class main {
abstract static class PStackComputableWithStep extends PStackComputable {
int step;
// phasing out the signature with subComputationResult
public final void step(VStack stack, Object subComputationResult) {
step((IPStack) stack);
}
void step(IPStack stack) {}
public String toString() { return shortClassName(this) + " step " + step; }
}
static boolean step(Steppable steppable) {
return steppable == null ? null : steppable.step();
}
static String shortClassName(Object o) {
if (o == null) return null;
Class c = o instanceof Class ? (Class) o : o.getClass();
String name = c.getName();
return shortenClassName(name);
}
static String shortenClassName(String name) {
if (name == null) return null;
int i = lastIndexOf(name, "$");
if (i < 0) i = lastIndexOf(name, ".");
return i < 0 ? name : substring(name, i+1);
}
static int lastIndexOf(String a, String b) {
return a == null || b == null ? -1 : a.lastIndexOf(b);
}
static int lastIndexOf(String a, char b) {
return a == null ? -1 : a.lastIndexOf(b);
}
// starts searching from i-1
static int lastIndexOf(List l, int i, A a) {
if (l == null) return -1;
for (i = min(l(l), i)-1; i >= 0; i--)
if (eq(l.get(i), a))
return i;
return -1;
}
static int lastIndexOf(List l, A a) {
if (l == null) return -1;
for (int i = l(l)-1; i >= 0; i--)
if (eq(l.get(i), a))
return i;
return -1;
}
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 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 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 int l(MultiSet ms) { return ms == null ? 0 : ms.size(); }
static int l(IMultiMap mm) { return mm == null ? 0 : mm.size(); }
static int l(AppendableChain a) { return a == null ? 0 : a.size; }
static boolean eq(Object a, Object b) {
return a == b || a != null && b != null && a.equals(b);
}
// a little kludge for stuff like eq(symbol, "$X")
static boolean eq(Symbol a, String b) {
return eq(str(a), b);
}
static int strL(String s) {
return s == null ? 0 : s.length();
}
static int lCharSequence(CharSequence s) {
return s == null ? 0 : s.length();
}
static int iteratorCount_int_close(Iterator 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 String str(Object o) {
return o == null ? "null" : o.toString();
}
static String str(char[] c) {
return new String(c);
}
static String str(char[] c, int offset, int count) {
return new String(c, offset, count);
}
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 void _handleError(Error e) {
//call(javax(), '_handleError, e);
}
interface IPStack extends IVStack {
// declare a probabilistic choice point
default void options(B function, IVF1... options) {
options(function, arrayIterator(options));
}
void options(B function, Iterable> options);
}
static class VStack implements Steppable, IVStack {
List stack = new ArrayList();
Object latestResult;
transient Object newResult;
// the null sentinel replaces a null function result
// when stored in latestResult
static class NullSentinel {}
static NullSentinel nullSentinel = new NullSentinel();
VStack() {}
VStack(Computable computation) { push(computation); }
VStack(Iterable l) { pushAll(l); }
// A is what the function returns
interface Computable {
public void step(VStack stack, Object subComputationResult);
}
private Object deSentinel(Object o) {
return o instanceof NullSentinel ? null : o;
}
private Object sentinel(Object o) {
return o == null ? nullSentinel : o;
}
// called by computations or users to start a subroutine
public void push(Computable computation) {
stack.add(assertNotNull(computation));
}
// perform a computation step. returns false iff done
public boolean step() {
if (empty(stack)) return false;
newResult = null;
last(stack).step(this, result());
latestResult = newResult;
newResult = null;
return true;
}
// called from a computation to return a value
public void _return(Object value) {
newResult = sentinel(value);
pop();
}
// called from a computation to tail-call another routine
final void replace(Computable computation){ tailCall(computation); }
void tailCall(Computable computation) {
pop();
stack.add(computation);
}
// all-in-one evaluation function - call on an empty stack
A compute(Computable computation) {
if (computation == null) return null;
push(computation);
stepAll(this);
return (A) latestResult;
}
// return result of just completed computation or sub-computation
final public Object subResult(){ return result(); }
public Object result() {
return deSentinel(latestResult);
}
boolean hasSubResult() { return latestResult != null; }
void pushAll(Iterable l) {
for (Computable c : unnullForIteration(l))
push(c);
}
void add(Runnable r) {
if (r == null) return;
push((stack, subComputationResult) -> r.run());
}
// caller of current function
Computable caller() {
return nextToLast(stack);
}
protected void pop() {
removeLast(stack);
}
boolean isEmpty() { return empty(stack); }
}
static interface Steppable {
public boolean step(); // return false if done
}
abstract static class PStackComputable implements VStack.Computable {
double probability = 1;
}
interface IVStack {
default void call(VStack.Computable f){ push(f); }
void push(VStack.Computable f);
default void _return() { _return(null); }
void _return(Object value);
Object subResult();
}
static interface IVF1 {
void get(A a);
}
static Concept options(List l) {
return l(l) == 1 ? first(l) : new Options(l);
}
static IterableIterator arrayIterator(A[] l) {
return l == null ? null : new IterableIterator() {
int i = 0;
public boolean hasNext() { return i < l.length; }
public A next() { return l[i++]; }
};
}
static A assertNotNull(A a) {
assertTrue(a != null);
return a;
}
static A assertNotNull(String msg, A a) {
assertTrue(msg, a != null);
return a;
}
static boolean empty(Collection c) { return c == null || c.isEmpty(); }
static boolean empty(Iterable c) { return c == null || !c.iterator().hasNext(); }
static boolean empty(CharSequence s) { return s == null || s.length() == 0; }
static boolean empty(Map map) { return map == null || map.isEmpty(); }
static boolean empty(Object[] o) { return o == null || o.length == 0; }
static boolean empty(BitSet bs) { return bs == null || bs.isEmpty(); }
static boolean empty(Object o) {
if (o instanceof Collection) return empty((Collection) o);
if (o instanceof String) return empty((String) o);
if (o instanceof Map) return empty((Map) o);
if (o instanceof Object[]) return empty((Object[]) o);
if (o instanceof byte[]) return empty((byte[]) o);
if (o == null) return true;
throw fail("unknown type for 'empty': " + getType(o));
}
static boolean empty(Iterator i) { return i == null || !i.hasNext(); }
static boolean empty(double[] a) { return a == null || a.length == 0; }
static boolean empty(float[] a) { return a == null || a.length == 0; }
static boolean empty(int[] a) { return a == null || a.length == 0; }
static boolean empty(long[] a) { return a == null || a.length == 0; }
static boolean empty(byte[] a) { return a == null || a.length == 0; }
static boolean empty(short[] a) { return a == null || a.length == 0; }
static boolean empty(MultiSet ms) { return ms == null || ms.isEmpty(); }
static boolean empty(IMultiMap mm) { return mm == null || mm.size() == 0; }
static boolean empty(File f) { return getFileSize(f) == 0; }
static boolean empty(Rect r) { return !(r != null && r.w != 0 && r.h != 0); }
static boolean empty(Chain c) { return c == null; }
static boolean empty(AppendableChain c) { return c == null; }
static A last(List l) {
return empty(l) ? null : l.get(l.size()-1);
}
static char last(String s) {
return empty(s) ? '#' : s.charAt(l(s)-1);
}
static byte last(byte[] a) {
return l(a) != 0 ? a[l(a)-1] : 0;
}
static int last(int[] a) {
return l(a) != 0 ? a[l(a)-1] : 0;
}
static double last(double[] a) {
return l(a) != 0 ? a[l(a)-1] : 0;
}
static A last(A[] a) {
return l(a) != 0 ? a[l(a)-1] : null;
}
static A last(Iterator it) {
A a = null;
while (it.hasNext()) { ping(); a = it.next(); }
return a;
}
static A last(Collection l) {
if (l == null) return null;
if (l instanceof List) return (A) last((List) l);
if (l instanceof SortedSet) return (A) last((SortedSet) l);
Iterator it = iterator(l);
A a = null;
while (it.hasNext()) { ping(); a = it.next(); }
return a;
}
static A last(SortedSet l) {
return l == null ? null : l.last();
}
static A last(ReverseChain l) {
return l == null ? null : l.element;
}
static A last(CompactLinkedHashSet set) {
return set == null ? null : set.last();
}
// returns number of steps
static long stepAll(Steppable s) {
long steps = 0;
if (s != null) {
var pingSource = pingSource();
while (true) {
ping(pingSource);
if (s.step())
++steps;
else
break;
}
}
return steps;
}
static String unnullForIteration(String s) {
return s == null ? "" : s;
}
static Collection unnullForIteration(Collection l) {
return l == null ? immutableEmptyList() : l;
}
static List unnullForIteration(List l) { return l == null ? immutableEmptyList() : l; }
static int[] unnullForIteration(int[] l) { return l == null ? emptyIntArray() : l; }
static char[] unnullForIteration(char[] l) { return l == null ? emptyCharArray() : l; }
static double[] unnullForIteration(double[] l) { return l == null ? emptyDoubleArray() : l; }
static short[] unnullForIteration(short[] l) { return l == null ? emptyShortArray() : l; }
static Map unnullForIteration(Map l) {
return l == null ? immutableEmptyMap() : l;
}
static Iterable unnullForIteration(Iterable i) {
return i == null ? immutableEmptyList() : i;
}
static A[] unnullForIteration(A[] a) {
return a == null ? (A[]) emptyObjectArray() : a;
}
static BitSet unnullForIteration(BitSet b) {
return b == null ? new BitSet() : b;
}
static Pt unnullForIteration(Pt p) {
return p == null ? new Pt() : p;
}
//ifclass Symbol
static Symbol unnullForIteration(Symbol s) {
return s == null ? emptySymbol() : s;
}
//endif
static Pair unnullForIteration(Pair p) {
return p != null ? p : new Pair(null, null);
}
static long unnullForIteration(Long l) { return l == null ? 0L : l; }
static A nextToLast(List l) {
return get(l, l(l)-2);
}
static void removeLast(List l) {
if (!l.isEmpty())
l.remove(l(l)-1);
}
static void removeLast(List l, int n) {
removeSubList(l, l(l)-n);
}
static void removeLast(int n, List l) {
removeLast(l, n);
}
static Object first(Object list) {
return first((Iterable) list);
}
static A first(List list) {
return empty(list) ? null : list.get(0);
}
static A first(A[] bla) {
return bla == null || bla.length == 0 ? null : bla[0];
}
static Pair first(Map map) {
return mapEntryToPair(first(entrySet(map)));
}
static Pair first(MultiMap mm) {
if (mm == null) return null;
var e = first(mm.data.entrySet());
if (e == null) return null;
return pair(e.getKey(), first(e.getValue()));
}
static A first(IterableIterator i) {
return first((Iterator) i);
}
static A first(Iterator i) {
return i == null || !i.hasNext() ? null : i.next();
}
static A first(Iterable i) {
if (i == null) return null;
Iterator it = i.iterator();
return it.hasNext() ? it.next() : null;
}
static Character first(String s) { return empty(s) ? null : s.charAt(0); }
static Character first(CharSequence s) { return empty(s) ? null : s.charAt(0); }
static A first(Pair p) {
return p == null ? null : p.a;
}
static A first(T3 t) {
return t == null ? null : t.a;
}
static Byte first(byte[] l) {
return empty(l) ? null : l[0];
}
static A first(A[] l, IF1 pred) {
return firstThat(l, pred);
}
static A first(Iterable l, IF1 pred) {
return firstThat(l, pred);
}
static A first(IF1 pred, Iterable l) {
return firstThat(pred, l);
}
static A first(AppendableChain a) {
return a == null ? null : a.element;
}
static void assertTrue(Object o) {
if (!(eq(o, true) /*|| isTrue(pcallF(o))*/))
throw fail(str(o));
}
static boolean assertTrue(String msg, boolean b) {
if (!b)
throw fail(msg);
return b;
}
static boolean assertTrue(boolean b) {
if (!b)
throw fail("oops");
return b;
}
static volatile StringBuffer local_log = new StringBuffer(); // not redirected
static boolean printAlsoToSystemOut = true;
static volatile Appendable print_log = local_log; // might be redirected, e.g. to main bot
// in bytes - will cut to half that
static volatile int print_log_max = 1024*1024;
static volatile int local_log_max = 100*1024;
static boolean print_silent = false; // total mute if set
static Object print_byThread_lock = new Object();
static volatile ThreadLocal