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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.*;
// draw faint one times one pixel chess board to indicate how big a pixel is
class main {
static void drawPixelChessboard(BufferedImage image) {
int w = image.getWidth(), h = image.getHeight();
for (int y = 0; y < h; y++)
for (int x = 0; x < h; x++)
setPixel(image, x, y, even(x+y) ? Color.lightGray : Color.white);
}
static void setPixel(BufferedImage img, int p_x, int p_y, Color color) { setPixel(img, pt(p_x, p_y), color); }
static void setPixel(BufferedImage img, Pt p, Color color) {
{ if (img != null) img.setRGB(p.x, p.y, colorToIntOpaque(color)); }
}
static boolean even(int i) {
return (i & 1) == 0;
}
static boolean even(long i) {
return (i & 1) == 0;
}
static boolean even(BigInteger n) {
return even(n.intValue());
}
static Pt pt(int x, int y) {
return new Pt(x, y);
}
static Pt pt(int x) {
return new Pt(x, x);
}
static int colorToIntOpaque(Color c) {
return c.getRGB() | 0xFF000000;
}
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; }
}
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 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);
}
}