Problem: Gerald got tired of playing board games with the usual six-sided die, and he bought a toy called Randomizer. It functions as follows.

A Randomizer has its own coordinate plane on which a strictly convex polygon is painted, the polygon is called a basic polygon. If you shake a Randomizer, it draws some nondegenerate (i.e. having a non-zero area) convex polygon with vertices at some vertices of the basic polygon. The result of the roll (more precisely, the result of the shaking) is considered to be the number of points with integer coordinates, which were strictly inside (the points on the border are not considered) the selected polygon. Now Gerald is wondering: what is the expected result of shaking the Randomizer?

During the shaking the Randomizer considers all the possible non-degenerate convex polygons with vertices at the vertices of the basic polygon. Let's assume that there are k versions of the polygons. Then the Randomizer chooses each of them with probability <image>.

Input

The first line of the input contains a single integer n (3 ≤ n ≤ 100 000) — the number of vertices of the basic polygon. 

Next n lines contain the coordinates of the vertices of the basic polygon. The i-th of these lines contain two integers xi and yi ( - 109 ≤ xi, yi ≤ 109) — the coordinates of the i-th vertex of the polygon. The vertices are given in the counter-clockwise order.

Output

Print the sought expected value with absolute or relative error at most 10 - 9.

Examples

Input

4
0 0
2 0
2 2
0 2


Output

0.2


Input

5
0 0
2 0
2 2
1 3
0 2


Output

0.8125

Note

A polygon is called strictly convex if it is convex and no its vertices lie on the same line.

Let's assume that a random variable takes values x1, ..., xn with probabilities p1, ..., pn, correspondingly. Then the expected value of this variable equals to <image>.

Solution (in JAVA):

Solution:
//package round313;
import java.io.ByteArrayInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.math.BigDecimal;
import java.math.BigInteger;
import java.util.Arrays;
import java.util.InputMismatchException;

public class D2 {
	InputStream is;
	PrintWriter out;
	String INPUT = "";
	
	long[] e2;
	
	void sl(long[] x)
	{
		x[1] *= 2;
		add(x, x[0]);
	}
	
	void add(long[] x, long a) {
		if(a < 0){
			if(x[0] + a < 0){
				x[0] += 1L<<63;
				x[0] += a;
				x[1]--;
			}else{
				x[0] += a;
			}
		}else{
			if(x[0] + a < 0){
				x[0] -= 1L<<63;
				x[0] += a;
				x[1]++;
			}else{
				x[0] += a;
			}
		}
	}
	
	void solve()
	{
		int n = ni();
		long[][] co = new long[n][];
		long minx = Long.MAX_VALUE;
		for(int i = 0;i < n;i++){
			co[i] = new long[]{ni(), ni()};
			minx = Math.min(minx, co[i][0]);
		}
		
		int D = 43;
		long O = -1000000001;
		long[] x = new long[2];
		long[] y = new long[2];
		double hn = Math.pow(0.5, n);
		for(int d = 1;d <= D && d < n;d++){
			for(int i = 0, j = d%n;i < n;i++,j++){
				if(j == n)j = 0;
				long ax = co[i][0], ay = co[i][1];
				long bx = co[j][0], by = co[j][1];
				if(bx < ax){
					long hl = by-O, hr = ay-O;
					long base = ax-bx;
					long inner = ((hl+hr)*base-(gcd(Math.abs(hr-hl), base)+hl+hr+base))/2 + 1;
					
					add(x, hl+inner-1);
					add(y, hl+inner-1);
				}else if(bx > ax){
					long hl = ay-O, hr = by-O;
					long base = bx-ax;
					long g = gcd(Math.abs(hr-hl), base);
					long inner = ((hl+hr)*base-(g+hl+hr+base))/2 + 1;
					
					add(x, -(hl+inner-1+g));
					add(y, -(hl+inner-1+g));
				}else{
					if(by < ay){
						add(x, -(ay-by));
						add(y, -(ay-by));
					}
				}
			}
			sl(x);
//			E += hd-hn;
		}
		BigInteger X = BigInteger.valueOf(x[1]).shiftLeft(63).add(BigInteger.valueOf(x[0]));
		BigInteger Y = BigInteger.valueOf(y[1]).shiftLeft(63).add(BigInteger.valueOf(y[0]));
		BigDecimal XX = new BigDecimal(X).divide(BigDecimal.valueOf(2L).pow(Math.min(D, n-1)+2));
		BigDecimal YY = new BigDecimal(Y).divide(BigDecimal.valueOf(2L).pow(Math.min(n, 100)));
		double E = XX.subtract(YY).doubleValue();
		
		E /= 1. - (1+n+(long)n*(n-1)/2)*hn;
		E += 1;

		out.printf("%.15f\n", E);
	}
	
	public static long gcd(long a, long b) {
		while (b > 0) {
			long c = a;
			a = b;
			b = c % b;
		}
		return a;
	}
	
	
	void run() throws Exception
	{
//		int n = 100000, m = 99999;
//		Random gen = new Random();
//		StringBuilder sb = new StringBuilder();
//		sb.append(n + " ");
//		for (int i = 0; i < n; i++) {
//			long r = 1000000000;
//			sb.append(Math.round(r*Math.cos(i*Math.PI*2/n)) + " ");
//			sb.append(Math.round(r*Math.sin(i*Math.PI*2/n)) + " ");
//		}
//		INPUT = sb.toString();
		
		is = oj ? System.in : new ByteArrayInputStream(INPUT.getBytes());
		out = new PrintWriter(System.out);
		
		long s = System.currentTimeMillis();
		solve();
		out.flush();
		tr(System.currentTimeMillis()-s+"ms");
	}
	
	public static void main(String[] args) throws Exception { new D2().run(); }
	
	private byte[] inbuf = new byte[1024];
	private int lenbuf = 0, ptrbuf = 0;
	
	private int readByte()
	{
		if(lenbuf == -1)throw new InputMismatchException();
		if(ptrbuf >= lenbuf){
			ptrbuf = 0;
			try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
			if(lenbuf <= 0)return -1;
		}
		return inbuf[ptrbuf++];
	}
	
	private boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
	private int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
	
	private double nd() { return Double.parseDouble(ns()); }
	private char nc() { return (char)skip(); }
	
	private String ns()
	{
		int b = skip();
		StringBuilder sb = new StringBuilder();
		while(!(isSpaceChar(b))){ // when nextLine, (isSpaceChar(b) && b != ' ')
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	
	private char[] ns(int n)
	{
		char[] buf = new char[n];
		int b = skip(), p = 0;
		while(p < n && !(isSpaceChar(b))){
			buf[p++] = (char)b;
			b = readByte();
		}
		return n == p ? buf : Arrays.copyOf(buf, p);
	}
	
	private char[][] nm(int n, int m)
	{
		char[][] map = new char[n][];
		for(int i = 0;i < n;i++)map[i] = ns(m);
		return map;
	}
	
	private int[] na(int n)
	{
		int[] a = new int[n];
		for(int i = 0;i < n;i++)a[i] = ni();
		return a;
	}
	
	private int ni()
	{
		int num = 0, b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private long nl()
	{
		long num = 0;
		int b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private boolean oj = System.getProperty("ONLINE_JUDGE") != null;
	private void tr(Object... o) { if(!oj)System.out.println(Arrays.deepToString(o)); }
}