Problem:
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Intersection of Circles

For given two circles $c1$ and $c2$, print

4

if they do not cross (there are 4 common tangent lines),

3

if they are circumscribed (there are 3 common tangent lines),

2

if they intersect (there are 2 common tangent lines),

1

if a circle is inscribed in another (there are 1 common tangent line),

0

if a circle includes another (there is no common tangent line).

Input

Coordinates and radii of $c1$ and $c2$ are given in the following format.

$c1x \; c1y \; c1r$

$c2x \; c2y \; c2r$

$c1x$, $c1y$ and $c1r$ represent the center coordinate and radius of the first circle.
$c2x$, $c2y$ and $c2r$ represent the center coordinate and radius of the second circle.
All input values are given in integers.

Output

Print "4", "3", "2", "1" or "0" in a line.

Constraints

$-1,000 \leq c1x, c1y, c2x, c2y \leq 1,000$

$1 \leq c1r, c2r \leq 1,000$

$c1$ and $c2$ are different

Sample Input 1

1 1 1
6 2 2

Sample Output 1

4

Sample Input 2

1 2 1
4 2 2

Sample Output 2

3

Sample Input 3

1 2 1
3 2 2

Sample Output 3

2

Sample Input 4

0 0 1
1 0 2

Sample Output 4

1

Sample Input 5

0 0 1
0 0 2

Sample Output 5

0

Solution in C# (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Collections;
using System.Collections.Specialized;
using System.Linq;
using System.Text;
using System.IO;
using System.Reflection;
using static System.Math;
using System.Numerics;
using static Exg;
static class Program{
	const int mod=(int)1e9+7;
	const double eps=1e-11;
	static void Main(){
		Sc sc=new Sc();
		var e1=sc.Da;
		var p1=new Pair(e1);
		var e2=sc.Da;
		var p2=new Pair(e2);
		var a=Gp11(p1,e1[2],p2,e2[2]);
		var ans=Gp0(p1,p2)<=Max(e1[2],e2[2]);
		Console.WriteLine("{0}",Gp0(p1,p2)>Max(e1[2],e2[2])?4-a.Length:a.Length);
	}
}

public struct Pair{
	public double x,y;
	public Pair(double x,double y){this.x=x;this.y=y;}
	public Pair(double[] da){x=da[0];y=da[1];}
	public Pair(Pair p){x=p.x;y=p.y;}
	public override string ToString()=>x.ToString()+" "+y.ToString();
	public static Pair operator+(Pair a,Pair b){a.x+=b.x;a.y+=b.y;return a;}
	public static Pair operator-(Pair a,Pair b){a.x-=b.x;a.y-=b.y;return a;}
	public static Pair operator*(Pair a,double n){a.x*=n;a.y*=n;return a;}
	public static Pair operator/(Pair a,double n){a.x/=n;a.y/=n;return a;}
}
static class Exg{
	static public double Gp0(Pair a,Pair b){return Sqrt((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y));}
	static public Tuple<bool,Pair> Gp1(Pair p1,Pair p2,Pair p3,Pair p4){
		double t1=(p4.y-p3.y)*(p4.x-p1.x)-(p4.x-p3.x)*(p4.y-p1.y);
		double t2=(p2.x-p1.x)*(p4.y-p1.y)-(p2.y-p1.y)*(p4.x-p1.x);
		double t3=(p2.x-p1.x)*(p4.y-p3.y)-(p2.y-p1.y)*(p4.x-p3.x);
		if(t3==0){
			if(Abs(Atan2(p2.y-p1.y,p2.x-p1.x))!=Abs(Atan2(p3.y-p1.y,p3.x-p1.x))){return Tuple.Create(p1.x==p3.x&&p1.y==p3.y?true:false,p1);}
			var a=new Tuple<Pair,int>[]{Tuple.Create(p1,1),Tuple.Create(p2,2),Tuple.Create(p3,3),Tuple.Create(p4,4)};
			Array.Sort(a,(d,e)=>{var c=d.Item1.x.CompareTo(e.Item1.y);return c==0?e.Item1.y.CompareTo(d.Item1.y):c;});
			return Tuple.Create((Abs(a[0].Item2-a[1].Item2)!=1||Abs(a[2].Item2-a[3].Item2)!=1)||(a[1].Item1.x==a[2].Item1.x&&a[1].Item1.y==a[2].Item1.y)?true:false,a[1].Item1);
		}
		double t4=t1/t3,t5=t2/t3;
		if(t4<0||t4>1||t5<0||t5>1){return Tuple.Create(false,new Pair(0,0));}
		else{return Tuple.Create(true,new Pair(p1.x+t4*(p2.x-p1.x),p1.y+t4*(p2.y-p1.y)));}
	}
	static public double Gp2(Pair p1,Pair p2,Pair p0){return Abs((p2.y-p1.y)*p0.x-(p2.x-p1.x)*p0.y+p2.x*p1.y-p1.x*p2.y)/Sqrt((p2.y-p1.y)*(p2.y-p1.y)+(p2.x-p1.x)*(p2.x-p1.x));}
	static public double Gp3(Pair p1,Pair p2,Pair p0){return (p1.x-p2.x)*(p0.y-p1.y)+(p1.y-p2.y)*(p1.x-p0.x);}
	static public double Gp4(Pair a,Pair b,Pair c){
		double a1=a.x-b.x,a2=a.y-b.y,c1=c.x-b.x,c2=c.y-b.y;
		return Acos((a1*c1+a2*c2)/Sqrt((a1*a1+a2*a2)*(c1*c1+c2*c2)))*180/PI;
	}
	static public Tuple<bool,Pair> Gp5(Pair p1,Pair p2,Pair p0){
		double rx=p0.x,ry=p0.y;
		if(p1.x==p2.x){rx=p1.x;}
		else if(p1.y==p2.y){ry=p1.y;}
		else{
			double m1=(p2.y-p1.y)/(p2.x-p1.x),b1=p1.y-(m1*p1.x),m2=-1.0/m1,b2=p0.y-(m2*p0.x);
			rx=(b2-b1)/(m1-m2);
			ry=(b2*m1-b1*m2)/(m1-m2);
		}
		return Tuple.Create(((p1.x>=rx&&p2.x<=rx)||(p2.x>=rx&&p1.x<=rx))&&((p1.y>=ry&&p2.y<=ry)||(p2.y>=ry&&p1.y<=ry)),new Pair(rx,ry));
	}
	static public Pair Gp6(Pair a,Pair b,Pair c,int n){
		double l1=Sqrt((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y));
		double l2=Sqrt((c.x-b.x)*(c.x-b.x)+(c.y-b.y)*(c.y-b.y));
		double ex=(a.x+(b.x+(c.x-b.x)*(l1/l2)))/2,ey=(a.y+(b.y+(c.y-b.y)*(l1/l2)))/2;
		ex+=(ex-b.x)*n;ey+=(ey-b.y)*n;
		return new Pair(ex,ey);
	}
	static public Pair Gp7(Pair a,double k,double p){return new Pair(a.x+k*Cos(p),a.y+k*Sin(p));}
	static public bool Gp8(Pair a,Pair b,Pair c){return (c.x-a.x)*(b.y-a.y)==(b.x-a.x)*(c.y-a.y);}
	static public Pair Gp9(Pair a,Pair b,Pair c){
		double d=a.x*a.x+a.y*a.y,e=b.x*b.x+b.y*b.y,f=c.x*c.x+c.y*c.y,g=2*(a.x*(b.y-c.y)+b.x*(c.y-a.y)+c.x*(a.y-b.y));
		return new Pair((d*(b.y-c.y)+e*(c.y-a.y)+f*(a.y-b.y))/g,(d*(c.x-b.x)+e*(a.x-c.x)+f*(b.x-a.x))/g);
	}
	static public Pair[] Gp10(Pair p0,double r,Pair p1,Pair p2){
		double a=p2.y-p1.y,b=p1.x-p2.x;
		double d=p0.x*-a+p0.y*-b+a*p1.x+b*p1.y;
		double g=a*a+b*b;
		double k=g*r*r-d*d;
		if(k<0){return new Pair[]{};}
		if(k==0){return new Pair[]{new Pair((a*d)/g+p0.x,(b*d)/g+p0.y)};}
		k=Sqrt(k);
		return new Pair[]{new Pair((a*d-b*k)/g+p0.x,(b*d+a*k)/g+p0.y),new Pair((a*d+b*k)/g+p0.x,(b*d-a*k)/g+p0.y)};
	}
	static public Pair[] Gp11(Pair p1,double r1,Pair p2,double r2){
		p2-=p1;
		double a=(p2.x*p2.x+p2.y*p2.y+r1*r1-r2*r2)/2;
		double g=p2.x*p2.x+p2.y*p2.y;
		double k=g*r1*r1-a*a;
		if(k<0){return new Pair[]{};}
		if(k==0){return new Pair[]{new Pair((p2.x*a)/g,(p2.y*a)/g)+p1};}
		k=Sqrt(k);
		return new Pair[]{new Pair((p2.x*a+p2.y*k)/g,(p2.y*a-p2.x*k)/g)+p1,new Pair((p2.x*a-p2.y*k)/g,(p2.y*a+p2.x*k)/g)+p1};
	}
	static public Tuple<Pair,double>[] Psort(Pair[] pa,Pair p){
		var arr=new Tuple<Pair,double>[pa.Length];
		for(int i = 0;i<pa.Length;i++) {arr[i]=Tuple.Create(pa[i],Atan2(pa[i].y-p.y,pa[i].x-p.x)*180/PI+180);}
		Array.Sort(arr,(u,v)=>u.Item2.CompareTo(v.Item2));
		return arr;
	}
	static public Tuple<Pair,double>[] Psort2(Pair[] pa,Pair p){
		int n=pa.Length;
		var arr=new Tuple<Pair,double>[n];
		for(int i = 0;i<n;i++) {arr[i]=Tuple.Create(pa[i],Atan2(pa[i].y-p.y,pa[i].x-p.x)*180/PI+180);}
		Array.Sort(arr,(u,v)=>u.Item2.CompareTo(v.Item2));
		Array.Resize(ref arr,n<<1);
		for(int i = 0;i<n;i++) {arr[i+n]=Tuple.Create(arr[i].Item1,arr[i].Item2+360);}
		return arr;
	}
}
public class Sc{
	public int I{get{return int.Parse(Console.ReadLine());}}
	public long L{get{return long.Parse(Console.ReadLine());}}
	public double D{get{return double.Parse(Console.ReadLine());}}
	public string S{get{return Console.ReadLine();}}
	public int[] Ia{get{return Array.ConvertAll(Console.ReadLine().Split(),int.Parse);}}
	public long[] La{get{return Array.ConvertAll(Console.ReadLine().Split(),long.Parse);}}
	public double[] Da{get{return Array.ConvertAll(Console.ReadLine().Split(),double.Parse);}}
	public string[] Sa{get{return Console.ReadLine().Split();}}
	public object[] Oa{get{return Console.ReadLine().Split();}}
	public int[] Ia2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),int.Parse);}}
	public int[] Ia3(int a){return Array.ConvertAll((a.ToString()+" "+Console.ReadLine()).Split(),int.Parse);}
	public int[] Ia3(bool a,int b,bool c,int d){return Array.ConvertAll(((a?b.ToString()+" ":"")+Console.ReadLine()+(c?" "+d.ToString():"")).Split(),int.Parse);}
	public long[] La2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),long.Parse);}}
	public long[] La3(int a){return Array.ConvertAll((a.ToString()+" "+Console.ReadLine()).Split(),long.Parse);}
	public long[] La3(bool a,int b,bool c,int d){return Array.ConvertAll(((a?b.ToString()+" ":"")+Console.ReadLine()+(c?" "+d.ToString():"")).Split(),long.Parse);}
	public double[] Da2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),double.Parse);}}
	public double[] Da3(int a){return Array.ConvertAll((a.ToString()+" "+Console.ReadLine()).Split(),double.Parse);}
	public double[] Da3(bool a,int b,bool c,int d){return Array.ConvertAll(((a?b.ToString()+" ":"")+Console.ReadLine()+(c?" "+d.ToString():"")).Split(),double.Parse);}
	public T[] Arr<T>(int n,Func<T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f();}return a;}
	public T[] Arr<T>(int n,Func<int,T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i);}return a;}
	public T[] Arr<T>(int n,Func<string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(Console.ReadLine().Split());}return a;}
	public T[] Arr<T>(int n,Func<int,string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i,Console.ReadLine().Split());}return a;}
}