Problem:
Score: 600 points

Problem Statement

E869120 is initially standing at the origin (0, 0) in a two-dimensional plane.

He has N engines, which can be used as follows:

When E869120 uses the i-th engine, his X- and Y-coordinate change by x_i and y_i, respectively. In other words, if E869120 uses the i-th engine from coordinates (X, Y), he will move to the coordinates (X + x_i, Y + y_i).

E869120 can use these engines in any order, but each engine can be used at most once. He may also choose not to use some of the engines.

He wants to go as far as possible from the origin.
Let (X, Y) be his final coordinates. Find the maximum possible value of \sqrt{X^2 + Y^2}, the distance from the origin.

Constraints

1 \leq N \leq 100

-1 \ 000 \ 000 \leq x_i \leq 1 \ 000 \ 000

-1 \ 000 \ 000 \leq y_i \leq 1 \ 000 \ 000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
x_2 y_2
: :
x_N y_N

Output

Print the maximum possible final distance from the origin, as a real value.
Your output is considered correct when the relative or absolute error from the true answer is at most 10^{-10}.

Sample Input 1

3
0 10
5 -5
-5 -5

Sample Output 1

10.000000000000000000000000000000000000000000000000

The final distance from the origin can be 10 if we use the engines in one of the following three ways:

Use Engine 1 to move to (0, 10).

Use Engine 2 to move to (5, -5), and then use Engine 3 to move to (0, -10).

Use Engine 3 to move to (-5, -5), and then use Engine 2 to move to (0, -10).

The distance cannot be greater than 10, so the maximum possible distance is 10.

Sample Input 2

5
1 1
1 0
0 1
-1 0
0 -1

Sample Output 2

2.828427124746190097603377448419396157139343750753

The maximum possible final distance is 2 \sqrt{2} = 2.82842....
One of the ways to achieve it is:

Use Engine 1 to move to (1, 1), and then use Engine 2 to move to (2, 1), and finally use Engine 3 to move to (2, 2).

Sample Input 3

5
1 1
2 2
3 3
4 4
5 5

Sample Output 3

21.213203435596425732025330863145471178545078130654

If we use all the engines in the order 1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 5, we will end up at (15, 15), with the distance 15 \sqrt{2} = 21.2132... from the origin.

Sample Input 4

3
0 0
0 1
1 0

Sample Output 4

1.414213562373095048801688724209698078569671875376

There can be useless engines with (x_i, y_i) = (0, 0).

Sample Input 5

1
90447 91000

Sample Output 5

128303.000000000000000000000000000000000000000000000000

Note that there can be only one engine.

Sample Input 6

2
96000 -72000
-72000 54000

Sample Output 6

120000.000000000000000000000000000000000000000000000000

There can be only two engines, too.

Sample Input 7

10
1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
17 18
19 20

Sample Output 7

148.660687473185055226120082139313966514489855137208

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

//using System.Numerics;
using CS=System.Runtime.CompilerServices;
//using System.Diagnostics;
using static System.Math;
using ll=System.Int64;
using static Contest_F.Lib_IO;
using static Contest_F.Lib_Minifunc;


namespace Contest_F {

public static class Entry
{

	public static void Main() {
		checked{
			long n;
			Lib_IO.rm(out n);
			P<double,double>[] xy;
			rl(n,out xy);

			P<double,double,ll>[] rt = new P<double, double,ll>[2*n];
			for (int i = 0; i < n; i++)
			{
				var r = Sqrt(xy[i].A*xy[i].A+xy[i].B*xy[i].B);
				var t = Atan2(xy[i].B,xy[i].A);
				rt[2*i]=new P<double, double,ll>(r,t,i);
				rt[2*i+1]=new P<double, double,ll>(r,t+PI*2.0,i);
			}
			rt.sort(2);

			int j = 0;
			P<double,double> sum = new P<double, double>(0,0);
			double ans=0.0;
			while(rt[j].B<rt[0].B+PI){
				sum.A+=xy[rt[j].C].A;
				sum.B+=xy[rt[j].C].B;
				j++;
				chmax(ref ans, Sqrt(sum.A*sum.A+sum.B*sum.B));
			}
			for (int i = 1; i < n; i++)
			{
				sum.A-=xy[rt[i-1].C].A;
				sum.B-=xy[rt[i-1].C].B;
				chmax(ref ans, Sqrt(sum.A*sum.A+sum.B*sum.B));
				while(rt[j].B<rt[i].B+PI){
					sum.A+=xy[rt[j].C].A;
					sum.B+=xy[rt[j].C].B;
					j++;
					chmax(ref ans, Sqrt(sum.A*sum.A+sum.B*sum.B));
				}
			}

			Lib_IO.wm(ans);
		}
	}















}



#region BaseModule


public static class Lib_Minifunc{
	public const long MOD1 = 1000000007; // 10^9+7
	public const double EPS1 = 1e-8;
	public const long INF1 = 1000000000000000;  // 10^15
	[CS.MethodImpl(256)] public static void swap1<T>(ref T a,ref T b) { T t = a; a = b; b = t; }
	[CS.MethodImpl(256)] public static void swap2<T>(ref T a,ref T b) where T : IComparable { if (a.CompareTo(b)==1) swap1(ref a, ref b); }  // b の方が小さければ交換

	[CS.MethodImpl(256)] public static bool chmin<T>(ref T a,T b) where T : IComparable { bool r; if (r = (a.CompareTo(b)== 1)) a = b; return r; }  // b の方が小さければ a を更新
	[CS.MethodImpl(256)] public static bool chmax<T>(ref T a,T b) where T : IComparable { bool r; if (r = (a.CompareTo(b)==-1)) a = b; return r; }  // b の方が大きければ a を更新
	[CS.MethodImpl(256)] public static bool inside(long i,long n) => (0<=i&&i<n);
	[CS.MethodImpl(256)] public static bool inside(long x,long y,long w,long h) => (inside(x,w)&&inside(y,h));
	[CS.MethodImpl(256)] public static T min<T>(params T[] a) where T : IComparable { T m = a[0]; for (int i=1;i<a.Length;i++) chmin(ref m, a[i]); return m; }
	[CS.MethodImpl(256)] public static T max<T>(params T[] a) where T : IComparable { T m = a[0]; for (int i=1;i<a.Length;i++) chmax(ref m, a[i]); return m; }
	[CS.MethodImpl(256)] public static long mod(long a,long m=MOD1) { var v = a%m; return (v<0?(m<0?v-m:v+m):v); }
	[CS.MethodImpl(256)] public static long ceil(long a,long b) => (a%b==0||(a<0&&0<b)||(0<a&&b<0)?a/b:a/b+1);	// 整数商の切り上げ
	[CS.MethodImpl(256)] public static long round(double a) => Convert.ToInt64(a);	// 四捨五入
	public static string YES(bool a) => a?"YES":"NO";
	public static string yes(bool a) => a?"Yes":"No";
	[CS.MethodImpl(256)] public static P<T,U> initp<T,U>(T a,U b) => new P<T,U>(a,b);
	[CS.MethodImpl(256)] public static P<T,U,V> initp<T,U,V>(T a,U b,V c) => new P<T,U,V>(a,b,c);
	[CS.MethodImpl(256)] public static P<T,U,V,W> initp<T,U,V,W>(T a,U b,V c,W d) => new P<T,U,V,W>(a,b,c,d);
	public static T[] initc<T>(long l1) where T : new() { var m = new T[l1]; for (int i=0;i<m.Length;i++) m[i] = new T(); return m; }	// new制約は遅い
	public static int[] range(long s, long c) => Enumerable.Range((int)s,(int)c).ToArray();
	[CS.MethodImpl(256)] public static T[] inita<T>(long l1,T v=default(T)) => Enumerable.Repeat(v,(int)l1).ToArray();
	public static T[][] inita<T>(long l1,long l2,T v=default(T)) { var m = new T[l1][]; for (int i=0;i<m.Length;i++) m[i] = inita(l2,v); return m; }
	public static void shift(this IList<int> l,int o) { for (int i = 0;i<l.Count;i++) l[i]+=o; }
	public static void shift(this IList<long> l,long o) { for (int i = 0;i<l.Count;i++) l[i]+=o; }
	[CS.MethodImpl(256)] public static bool initd<T,U>(this IDictionary<T,U> d, T k, U v=default(U)) { if(d.ContainsKey(k)) { return false; } else { d[k] = v; return true; } }
	public static void sort<T>(this T[] a,int o=1, Comparison<T> f=null) where T : IComparable {
		switch(o){
			case  0: Array.Sort(a,f);break;
			case  1: Array.Sort(a,(x,y)=>x.CompareTo(y));break;
			case -1: Array.Sort(a,(x,y)=>y.CompareTo(x));break;
			default: break;
		}
	}
	public static void sort<T,U>(this P<T,U>[] a,int o, Comparison<P<T,U>> f=null) where T : IComparable where U : IComparable {
		switch(o){
			case  0: Array.Sort(a,f);break;
			case  1: Array.Sort(a,(x,y)=>x.A.CompareTo(y.A));break;
			case -1: Array.Sort(a,(x,y)=>y.A.CompareTo(x.A));break;
			case  2: Array.Sort(a,(x,y)=>x.B.CompareTo(y.B));break;
			case -2: Array.Sort(a,(x,y)=>y.B.CompareTo(x.B));break;
			default: break;
		}
	}
	public static void sort<T,U,V>(this P<T,U,V>[] a,int o, Comparison<P<T,U,V>> f=null) where T : IComparable where U : IComparable where V : IComparable {
		switch(o){
			case  0: Array.Sort(a,f);break;
			case  1: Array.Sort(a,(x,y)=>x.A.CompareTo(y.A));break;
			case -1: Array.Sort(a,(x,y)=>y.A.CompareTo(x.A));break;
			case  2: Array.Sort(a,(x,y)=>x.B.CompareTo(y.B));break;
			case -2: Array.Sort(a,(x,y)=>y.B.CompareTo(x.B));break;
			case  3: Array.Sort(a,(x,y)=>x.C.CompareTo(y.C));break;
			case -3: Array.Sort(a,(x,y)=>y.C.CompareTo(x.C));break;
			default: break;
		}
	}
	public static void sort<T,U,V,W>(this P<T,U,V,W>[] a,int o, Comparison<P<T,U,V,W>> f=null) where T : IComparable where U : IComparable where V : IComparable where W : IComparable {
		switch(o){
			case  0: Array.Sort(a,f);break;
			case  1: Array.Sort(a,(x,y)=>x.A.CompareTo(y.A));break;
			case -1: Array.Sort(a,(x,y)=>y.A.CompareTo(x.A));break;
			case  2: Array.Sort(a,(x,y)=>x.B.CompareTo(y.B));break;
			case -2: Array.Sort(a,(x,y)=>y.B.CompareTo(x.B));break;
			case  3: Array.Sort(a,(x,y)=>x.C.CompareTo(y.C));break;
			case -3: Array.Sort(a,(x,y)=>y.C.CompareTo(x.C));break;
			case  4: Array.Sort(a,(x,y)=>x.D.CompareTo(y.D));break;
			case -4: Array.Sort(a,(x,y)=>y.D.CompareTo(x.D));break;
			default: break;
		}
	}
}


public struct P<T,U>
{
	public T A; public U B;
	public P(T a,U b) { A = a; B = b; }
	public static implicit operator KeyValuePair<T,U>(P<T,U> a) => new KeyValuePair<T,U>(a.A,a.B);
	public static implicit operator P<T,U>(KeyValuePair<T,U> a) => new P<T,U>(a.Key,a.Value);
}
public struct P<T,U,V>
{
	public T A; public U B; public V C;
	public P(T a,U b,V c) { A = a; B = b; C = c; }
}
public struct P<T,U,V,W>
{
	public T A; public U B; public V C; public W D;
	public P(T a,U b,V c,W d) { A = a; B = b; C = c; D = d; }
}

public static class Lib_IO
{
	class Prt : System.IO.StreamWriter
	{
		public override IFormatProvider FormatProvider { get { return System.Globalization.CultureInfo.InvariantCulture; } }
		public Prt(System.IO.Stream s) : base(s,new UTF8Encoding(false,true)) {}
		public Prt(System.IO.Stream s,Encoding e) : base(s,e) {}
	}
	static Prt sw = new Prt(Console.OpenStandardOutput());
	static char[] sp = new char[] {' '};
	[CS.MethodImpl(256)] static bool eq<T,U>() => typeof(T).Equals(typeof(U));
	[CS.MethodImpl(256)] static T ct<T,U>(U a) => (T)Convert.ChangeType(a,typeof(T));
	[CS.MethodImpl(256)] static T cv<T>(string s) =>
		eq<T,int>()	? ct<T,int>(int.Parse(s))
	  : eq<T,long>()   ? ct<T,long>(long.Parse(s))
	  : eq<T,double>() ? ct<T,double>(double.Parse(s,System.Globalization.CultureInfo.InvariantCulture))
	  : eq<T,char>()   ? ct<T,char>(s[0])
					   : ct<T,string>(s);
	public static string[] rm<T>(out T a) {
		var z = Console.ReadLine().Split(sp,StringSplitOptions.RemoveEmptyEntries); 
		a = cv<T>(z[0]);
		return z;
	}
	public static string[] rm<T,U>(out T a,out U b) {
		var z = rm<T>(out a); 
		b = cv<U>(z[1]);
		return z;
	}
	public static string[] rm<T,U,V>(out T a,out U b,out V c) {
		var z = rm<T,U>(out a,out b); 
		c = cv<V>(z[2]);
		return z;
	}
	public static string[] rm<T,U,V,W>(out T a,out U b,out V c,out W d) {
		var z = rm<T,U,V>(out a,out b,out c); 
		d = cv<W>(z[3]);
		return z;
	}
	public static string[] rm<T,U,V,W,X>(out T a,out U b,out V c,out W d,out X e) {
		var z = rm<T,U,V,W>(out a,out b,out c,out d); 
		e = cv<X>(z[4]);
		return z;
	}
	public static string[] rm<T,U,V,W,X,Y>(out T a,out U b,out V c,out W d,out X e,out Y f) {
		var z = rm<T,U,V,W,X>(out a,out b,out c,out d,out e); 
		f = cv<Y>(z[5]);
		return z;
	}
	public static string[] ra<T>(out T[] a) {		
		var z = Console.ReadLine().Split(sp,StringSplitOptions.RemoveEmptyEntries);
		a = new T[z.Length];
		for(int i=0;i<a.Length;i++) a[i] = cv<T>(z[i]);
		return z;
	}
	public static string[][] rl<T>(long n,out T[] a) {
		a = new T[n];
		var y = new string[n][];
		for(int i=0;i<n;i++) y[i] = rm(out a[i]);
		return y;
	}
	public static string[][] rl<T,U>(long n,out P<T,U>[] a) {
		a = new P<T,U>[n];
		var y = new string[n][];
		for(int i=0;i<n;i++) {
			T o; U p;
			y[i] = rm(out o,out p);
			a[i] = new P<T,U>(o,p);
		}
		return y;
	}
	public static string[][] rl<T,U,V>(long n,out P<T,U,V>[] a) {
		a = new P<T,U,V>[n];
		var y = new string[n][];
		for(int i=0;i<n;i++) {
			T o; U p; V q;
			y[i] = rm(out o,out p,out q);
			a[i] = new P<T,U,V>(o,p,q);
		}
		return y;
	}
	public static string[][] rl<T,U,V,W>(long n,out P<T,U,V,W>[] a) {
		a = new P<T,U,V,W>[n];
		var y = new string[n][];
		for(int i=0;i<n;i++) {
			T o; U p; V q; W r;
			y[i] = rm(out o,out p,out q,out r);
			a[i] = new P<T,U,V,W>(o,p,q,r);
		}
		return y;
	}
	public static string[][] rx<T>(long n,out T[][] a) {
		a = new T[n][];
		var y = new string[n][];
		for(int i=0;i<n;i++) y[i] = ra(out a[i]);
		return y;
	}

	static void wwp(Action act){
		sw.AutoFlush = false;
		Console.SetOut(sw);
		act();
		Console.Out.Flush();
		sw.AutoFlush = true;
		Console.SetOut(sw);
	}
	[CS.MethodImpl(256)] static string wfm(Type t) =>t.Equals(typeof(double)) ? "{0:F10}" : "{0}";
	public static void wm(params object[] a) {
		wwp(()=>{
			for(int i=0;i<a.Length-1;i++) Console.Write(wfm(a[i].GetType())+" ",a[i]);
			Console.WriteLine(wfm(a[a.Length-1].GetType()),a[a.Length-1]);
		});
	}
	public static void wa<T>(IList<T> a) { wxa(new IList<T>[]{a}, " "); }
	public static void wl<T>(IList<T> a) {
		wwp(()=>{
			var f = wfm(typeof(T));
			if(a.Count==0){
				Console.WriteLine();
			}else{
				foreach(T x in a) Console.WriteLine(f,x);
			}
		});
	}
	static void wxa<T>(IList<IList<T>> a, string s) {
		wwp(()=>{
			var f = wfm(typeof(T));
			var g = f + s;
			foreach(var b in a) {
				if(b.Count==0){
					Console.WriteLine();
				}else{
					for(int i=0;i<b.Count-1;i++) Console.Write(g,b[i]);
					Console.WriteLine(f,b[b.Count-1]);
				}
			}
		});
	}
	public static void wx<T>(IList<IList<T>> a) { wxa(a, " "); }
	public static void wg<T>(IList<IList<T>> a) { wxa(a, ""); }
}

#endregion



}