Problem: A crop circle suddenly appeared on the vast agricultural land of Argentina. A total of n crop circles were confirmed, including overlapping, popping, large, and small ones.

When a mystery hunter tried to capture the whole picture of a crop circle from the air, he found it difficult to show the beautiful pattern in the image because the outline of each circle was not clear.

Therefore, the mystery hunter proposed to emphasize the contour by installing a string of special material along the contour.

The part where the string is installed is the part on the circumference of one crop circle that is not included in any other crop circle.

The film crew asked you to create a program to measure the required string length. Enter the center coordinates and radius of each crop circle and create a program to report the length of the string to be installed.

For reference, the first and second cases of the input / output examples are shown in FIGS. 1 and 2, respectively. The part of the string is shown by a thick line.


<image>


Figure 1



<image>


Figure 2



Constraints

* n ≤ 100
* -1000 ≤ xi, yi ≤ 1000
* 0 <ri ≤ 100

Input

Multiple datasets are given as input. Each dataset is given in the following format:

n (number of crop circles: integer)
x1 y1 r1 (center coordinates and radius of the first circle: real numbers separated by blanks)
x2 y2 r2 (center coordinates and radius of the second circle: real numbers separated by blanks)
..
..
xn yn rn (center coordinates and radius of the nth circle: real numbers separated by blanks)


When n is 0, it indicates the end of input.

Output

Output the string length on one line for each dataset. The output may contain an error of 0.000001 or less.

Example

Input

4
6 4 2
4 7 3
7 10 3
13 6 1
4
7 4 2
4 7 3
7 10 3
11 6 3
0


Output

39.664699289572
45.627024663706

Solution (in JAVA):

Solution:
import java.util.ArrayList;
import java.util.List;
import java.util.Scanner;

//Crop Circle
public class Main{

	final double EPS = 1e-8;
	final double[] BASE = {1, 0};
	
	double dot(double[] a, double[] b){
		return a[0]*b[0]+a[1]*b[1];
	}
	double cross(double[] a, double[] b){
		return a[0]*b[1]-a[1]*b[0];
	}
	double norm(double[] a){
		return Math.sqrt(a[0]*a[0]+a[1]*a[1]);
	}
	double[] sub(double[] a, double[] b){
		return new double[]{a[0]-b[0], a[1]-b[1]};
	}
	void swap(double[] a, double[] b){
		double t = a[0];
		a[0] = b[0]; b[0] = t;
		t = a[1];
		a[1] = b[1]; b[1] = t;
	}
	double angleTan(double[] a, double[] b){
		return Math.atan2(cross(a, b), dot(a, b));
	}
	
	int checkCircleOverlap(double x1, double y1, double r1, double x2, double y2, double r2){
		double d = (x1-x2)*(x1-x2)+(y1-y2)*(y1-y2), r = (r1-r2)*(r1-r2), R = (r1+r2)*(r1+r2);
		if(R < d)return 0;
		if(Math.abs(d-R)<EPS)return 1;
		if(r < d && d < R)return 2;
		if(Math.abs(d-r)<EPS)return -1;
		return -2;
	}
	
	double[][] circleCrossPoint(double x1, double y1, double r1, double x2, double y2, double r2){
		double vx = x2-x1, vy = y2-y1, D = Math.sqrt(vx*vx+vy*vy);
		vx/=D; vy/=D;
		vx*=r1; vy*=r1;
		double thita = Math.acos((r1*r1+D*D-r2*r2)/(2*r1*D));
		double px = Math.cos(thita)*vx-Math.sin(thita)*vy, py = Math.sin(thita)*vx+Math.cos(thita)*vy;
		double px2 = Math.cos(-thita)*vx-Math.sin(-thita)*vy, py2 = Math.sin(-thita)*vx+Math.cos(-thita)*vy;
		double[][] res = new double[2][2];
		res[0][0] = x1+px; res[0][1] = y1+py;
		res[1][0] = x1+px2; res[1][1] = y1+py2;
		if(ccw(new double[]{x1, y1}, new double[]{x2, y2}, res[0])==1)swap(res[0], res[1]);
		return res;
	}
	
	int ccw(double[] a, double[] b, double[] c){
		b = sub(b, a); c = sub(c, a);
		if(cross(b, c) > 0)return 1;
		if(cross(b, c) < 0)return -1;
		if(dot(b, c) < 0)return +2;
		if(norm(b) < norm(c))return -2;
		return 0;
	}
	
	class CircumferenceRange{
		List<double[]> ranges;
		public CircumferenceRange() {
			ranges = new ArrayList<double[]>();
			ranges.add(new double[]{0, 2*Math.PI});
		}
		private double trans(double thita){
			return thita<0?2*Math.PI+thita:thita;
		}
		void sub(double from, double to){
			double f = trans(from), t = trans(to);
			if(t < f){
				subOffer(0, t);
				subOffer(f, 2*Math.PI);
			}
			else subOffer(f, t);
		}
		private void subOffer(double from, double to){
			for(int i=0;i<ranges.size();i++){
				double[] d = ranges.get(i);
				if(to < d[0] || d[1] < from)continue;
				if(from < d[0]+EPS && d[1] < to+EPS){
					ranges.remove(i--);
				}
				else if(d[0] < from && to < d[1]){
					ranges.remove(i);
					ranges.add(i, new double[]{to, d[1]});
					ranges.add(i++, new double[]{d[0], from});
				}
				else if(to < d[1]){
					ranges.remove(i);
					ranges.add(i, new double[]{to, d[1]});
				}
				else {
					ranges.remove(i);
					ranges.add(i, new double[]{d[0], from});
				}
			}
		}
	}
	
	void run(){
		Scanner sc = new Scanner(System.in);
		for(;;){
			int n = sc.nextInt();
			if(n==0)break;
			double[] x = new double[n], y = new double[n], r = new double[n];
			CircumferenceRange[] cr = new CircumferenceRange[n];
			for(int i=0;i<n;i++){
				x[i] = sc.nextDouble(); y[i] = sc.nextDouble(); r[i] = sc.nextDouble();
				cr[i] = new CircumferenceRange();
			}
			for(int i=0;i<n;i++)for(int j=0;j<n;j++)if(i!=j){
				int c = checkCircleOverlap(x[i], y[i], r[i], x[j], y[j], r[j]);
				if(c<0 && r[j]<r[i]){
					cr[j].ranges.clear();
				}
				if(c!=2)continue;
				double[][] cp = circleCrossPoint(x[i], y[i], r[i], x[j], y[j], r[j]);
				cr[i].sub(angleTan(BASE, sub(cp[0], new double[]{x[i], y[i]})), angleTan(BASE, sub(cp[1], new double[]{x[i], y[i]})));
			}
			double res = 0;
			for(int i=0;i<n;i++)for(double[]d:cr[i].ranges)res+=r[i]*(d[1]-d[0]);
			System.out.printf("%.9f\n", res);
		}
	}
	
	public static void main(String[] args) {
		new Main().run();
	}
}