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;
import java.awt.geom.*;
import static x30_pkg.x30_util.DynamicObject;
import java.text.*;
import java.text.NumberFormat;
import java.util.TimeZone;
class main {
static double bitsPerPixel(byte[] compressed, BufferedImage originalImage) {
return doubleRatio(l(compressed)*8.0, area(originalImage));
}
static double doubleRatio(double x, double y) {
return y == 0 ? 0 : x/y;
}
static double doubleRatio(Seconds x, Seconds y) {
return doubleRatio(x.get(), y.get());
}
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(IMultiMap mm) { return mm == null ? 0 : mm.size(); }
static int l(IntSize o) { return o == null ? 0 : o.size(); }
static int area(Rect r) {
return rectArea(r);
}
static double area(DoubleRect r) {
return r == null ? 0 : r.w*r.h;
}
static int area(WidthAndHeight img) {
return img == null ? 0 : img.getWidth()*img.getHeight();
}
static int area(BufferedImage img) {
return img == null ? 0 : img.getWidth()*img.getHeight();
}
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 int rectArea(Rect r) {
return r == null ? 0 : r.w*r.h;
}
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);
}
static class Rect implements WidthAndHeight , IFieldsToList{
static final String _fieldOrder = "x y w h";
int x;
int y;
int w;
int h;
Rect() {}
Rect(int x, int y, int w, int h) {
this.h = h;
this.w = w;
this.y = y;
this.x = x;}
public boolean equals(Object o) {
if (!(o instanceof Rect)) return false;
Rect __1 = (Rect) o;
return x == __1.x && y == __1.y && w == __1.w && h == __1.h;
}
public int hashCode() {
int h = 2543108;
h = boostHashCombine(h, _hashCode(x));
h = boostHashCombine(h, _hashCode(y));
h = boostHashCombine(h, _hashCode(w));
h = boostHashCombine(h, _hashCode(h));
return h;
}
public Object[] _fieldsToList() { return new Object[] {x, y, w, h}; }
Rect(Rectangle r) {
x = r.x;
y = r.y;
w = r.width;
h = r.height;
}
Rect(Pt p, int w, int h) {
this.h = h;
this.w = w; x = p.x; y = p.y; }
Rect(Rect r) { x = r.x; y = r.y; w = r.w; h = r.h; }
final Rectangle getRectangle() {
return new Rectangle(x, y, w, h);
}
public String toString() {
return x + "," + y + " / " + w + "," + h;
}
final int x1() { return x; }
final int y1() { return y; }
final int x2() { return x + w; }
final int y2() { return y + h; }
final boolean contains(Pt p) {
return contains(p.x, p.y);
}
final boolean contains(int _x, int _y) {
return _x >= x && _y >= y && _x < x+w && _y < y+h;
}
final boolean contains(Rectangle r) {
return rectContains(this, r);
}
final boolean empty() { return w <= 0 || h <= 0; }
final public int getWidth() { return w; }
final public int getHeight() { return h; }
final public int area() { return w*h; }
WidthAndHeight widthAndHeight() { return main.widthAndHeight(w, h); }
}
static class DoubleRect {
double x, y, w, h;
DoubleRect() {}
DoubleRect(Rectangle r) {
x = r.x;
y = r.y;
w = r.width;
h = r.height;
}
DoubleRect(double x, double y, double w, double h) {
this.h = h;
this.w = w;
this.y = y;
this.x = x;}
// Huh. not implementing equals()/hashCode? Stefan is mysterious
boolean eq(Object o) {
if (!(o instanceof DoubleRect)) return false;
if (o == this) return true;
DoubleRect r = (DoubleRect) o;
return x == r.x && y == r.y && w == r.w && h == r.h;
}
public String toString() {
return x + "," + y + " / " + w + "," + h;
}
double x1() { return x; }
double y1() { return y; }
double x2() { return x + w; }
double y2() { return y + h; }
boolean contains(Pt p) {
return contains(p.x, p.y);
}
boolean contains(double _x, double _y) {
return _x >= x && _y >= y && _x < x+w && _y < y+h;
}
boolean empty() { return w <= 0 || h <= 0; }
}
static interface IFieldsToList {
Object[] _fieldsToList();
}
static class Pt implements Comparable, IDoublePt {
int x, y;
Pt() {}
Pt(Point p) {
x = p.x;
y = p.y;
}
Pt(int x, int y) {
this.y = y;
this.x = x;}
Point getPoint() {
return new Point(x, y);
}
public boolean equals(Object o) {
return o instanceof Pt && x == ((Pt) o).x && y == ((Pt) o).y;
}
public int hashCode() {
return boostHashCombine(x, y);
}
// compare in scan order
public int compareTo(Pt p) {
if (y != p.y) return cmp(y, p.y);
return cmp(x, p.x);
}
public String toString() {
return x + ", " + y;
}
double length() { return sqrt(x*x+y*y); }
public Pt minus(Pt p) { return ptMinus(this, p); }
public double x_double() { return x; }
public double y_double() { return y; }
}
static interface WidthAndHeight {
default int w(){ return getWidth(); }
default int width(){ return getWidth(); }
int getWidth();
default int h(){ return getHeight(); }
default int height(){ return getHeight(); }
int getHeight();
public default Rect bounds() { return rect(0, 0, getWidth(), getHeight()); }
default int area() { return toInt(areaAsLong()); }
default long areaAsLong() { return longMul(w(), h()); }
}
interface IDoublePt {
public double x_double();
public double y_double();
}
static int boostHashCombine(int a, int b) {
return a ^ (b + 0x9e3779b9 + (a << 6) + (a >>> 2));
// OLD (changed) 2022/3/10: ret a ^ (b + 0x9e3779b9 + (a << 6) + (a >> 2));
}
static int _hashCode(Object a) {
return a == null ? 0 : a.hashCode();
}
static boolean contains(Collection c, Object o) {
return c != null && c.contains(o);
}
static boolean contains(Iterable it, Object a) {
if (it != null)
for (Object o : it)
if (eq(a, o))
return true;
return false;
}
static boolean contains(Object[] x, Object o) {
if (x != null)
for (Object a : x)
if (eq(a, o))
return true;
return false;
}
static boolean contains(String s, char c) {
return s != null && s.indexOf(c) >= 0;
}
static boolean contains(String s, String b) {
return s != null && s.indexOf(b) >= 0;
}
static boolean contains(BitSet bs, int i) {
return bs != null && bs.get(i);
}
static boolean contains(Rect r, Pt p) { return rectContains(r, p); }
static boolean rectContains(int x1, int y1, int w, int h, Pt p) {
return p.x >= x1 && p.y >= y1 && p.x < x1+w && p.y < y1+h;
}
static boolean rectContains(Rect a, Rect b) {
return b.x >= a.x && b.y >= a.y && b.x2() <= a.x2() && b.y2() <= a.y2();
}
static boolean rectContains(Rect a, Rectangle b) {
return rectContains(a, toRect(b));
}
static boolean rectContains(Rect a, int x, int y) {
return a != null && a.contains(x, y);
}
static boolean rectContains(Rect a, Pt p) {
return a != null && p != null && a.contains(p);
}
static WidthAndHeight widthAndHeight(BufferedImage image) {
return image == null ? null : widthAndHeight(image.getWidth(), image.getHeight());
}
static WidthAndHeightImpl widthAndHeight(int w) { return widthAndHeight(w, w); }
static WidthAndHeightImpl widthAndHeight(int w, int h) {
return new WidthAndHeightImpl(w, h);
}
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);
}
static double sqrt(double x) {
return Math.sqrt(x);
}
static Pt ptMinus(Pt a, Pt b) {
if (b == null) return a;
return new Pt(a.x-b.x, a.y-b.y);
}
static int getWidth(Component c) {
return c == null ? 0 : (int) swingCall(c, "getWidth");
}
static int getHeight(Component c) {
return c == null ? 0 : (int) swingCall(c, "getHeight");
}
static Rect rect(int x, int y, int w, int h) {
return new Rect(x, y, w, h);
}
static Rect rect(Pt p, int w, int h) {
return new Rect(p.x, p.y, w, h);
}
static Rect rect(int w, int h) {
return new Rect(0, 0, w, h);
}
static int toInt(Object o) {
if (o == null) return 0;
if (o instanceof Number)
return ((Number) o).intValue();
if (o instanceof String)
return parseInt((String) o);
if (o instanceof Boolean)
return boolToInt((Boolean) o);
throw fail("woot not int: " + getClassName(o));
}
static int toInt(long l) {
if (l != (int) l) throw fail("Too large for int: " + l);
return (int) l;
}
static long longMul(long a, long b) {
return a*b;
}
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 Rect toRect(Rectangle r) {
return r == null ? null : new Rect(r);
}
static Rect toRect(RectangularShape r) {
return r == null ? null : toRect(r.getBounds());
}
static Rect toRect(DoubleRect r) {
if (r == null) return null;
int x = iround(r.x), y = iround(r.y);
return new Rect(x, y, iround(r.x2())-x, iround(r.y2())-y);
}
static Rect toRect(Rect r) { return r; }
static Object swingCall(final Object o, final String method, final Object... args) {
return swing(new F0