You are given a convex polygon $P$ with vertices $P_0, P_1, \ldots, P_{n-1}$, each having integer coordinates. On each edge $P_{i} P_{(i+1) \% n}$ of the polygon, choose a point $R_i$ uniformly at random. What is the expected area of the convex hull of these $n$ chosen points $R_0, R_1, \ldots R_{n-1}$ ?

-----Note-----
- Consider the area of the convex hull as zero if it contains less than 3 vertices.
- All the points $R_i$ are chosen independently of each other.
- Your answer is considered correct if and only if its absolute or relative error doesn't exceed $10^{-6}$.

-----Input-----
- The first line contains $n$, the number of vertices in the convex polygon.
- The next $n$ lines contain the coordinates of the vertices of the polygon in anti-clockwise order. 

-----Output-----
For each testcase, print the expected area of the convex hull of the $n$ randomly chosen points.

-----Constraints-----
- $3 \leq n \leq 10^5$
- The absolute values of all the coordinates $\leq 10^7$.
- All the points in the input are distinct.
- The described polygon $P$ is convex and the vertices of the polygon are given in anti-clockwise order. Also, no three vertices of the polygon are collinear.

-----Example Input-----
3
0 0
1 0
0 1

-----Example Output-----
0.1250000000

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

ns = {"__name__": "__main__"}
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = '# cook your dish here\ndef func(a,n):\n if n < 3:\n  return 0\n res_arr = []\n for i in range(n):\n  x = (a[i][0] + a[(i + 1) % n][0]) / 2\n  y = (a[i][1] + a[(i + 1) % n][1]) / 2\n  res_arr.append((x, y))\n l = len(res_arr)\n s = 0\n for i in range(n):\n  u = res_arr[i][0]*res_arr[(i+1) % l][1]\n  v = res_arr[i][1]*res_arr[(i+1) % l][0]\n  s += (u-v)\n return abs(s)/2\n\ndef __starting_point():\n n = int(input())\n arr = []\n for i in range(n):\n  x, y = map(int, input().split())\n  arr.append((x,y))\n \n res = func(arr, n)\n print(res)\n__starting_point()'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 13575