#include <bits/stdc++.h>
using namespace std;
using ll = long long;
#define rep(i,n) for(int (i)=0;(i)<(int)(n);++(i))
#define all(x) (x).begin(),(x).end()
#define pb push_back
#define fi first
#define se second
#define dbg(x) cout<<#x" = "<<((x))<<endl
template<class T,class U> ostream& operator<<(ostream& o, const pair<T,U> &p){o<<"("<<p.fi<<","<<p.se<<")";return o;}
template<class T> ostream& operator<<(ostream& o, const vector<T> &v){o<<"[";for(T t:v){o<<t<<",";}o<<"]";return o;}

#define double long double

/* 基本要素 */
typedef complex<double> Point;
typedef pair<Point, Point> Line;
typedef vector<Point> VP;
const double PI = acos(-1);
const double EPS = 1e-6; // 許容誤差^2
const double INF = 1e9;
#define X real()
#define Y imag()
#define LE(n,m) ((n) - (m) < EPS)
#define GE(n,m) (EPS > (m) - (n))
#define EQ(n,m) (abs((n)-(m)) < EPS)

// 内積　dot(a,b) = |a||b|cosθ
double dot(Point a, Point b) {
  return a.X*b.X + a.Y*b.Y;
}

// 外積　cross(a,b) = |a||b|sinθ
double cross(Point a, Point b) {
  return a.X*b.Y - a.Y*b.X;
}

// 点の進行方向
// !!! 誤差に注意 !!! (掛け算したものとかなり小さいものを比べているので)
int ccw(Point a, Point b, Point c) {
  b -= a;  c -= a;
  if (cross(b,c) >  EPS) return +1;  // counter clockwise
  if (cross(b,c) < -EPS) return -1;  // clockwise
  if (dot(b,c)   < -EPS) return +2;  // c--a--b on line
  if (norm(b) < norm(c)) return -2;  // a--b--c on line or a==b
  return 0;                          // a--c--b on line or a==c or b==c
}

// 2直線の交点
Point crosspointLL(Point a1, Point a2, Point b1, Point b2) {
  double d1 = cross(b2-b1, b1-a1);
  double d2 = cross(b2-b1, a2-a1);
  if (EQ(d1, 0) && EQ(d2, 0)) return a1;  // same line
  assert(!EQ(d2, 0)); // 交点がない
  return a1 + d1/d2 * (a2-a1);
}

using Vector = Point;

Point READ(){
    double x,y;
    cin >>x >>y;
    return Point(x,y);
}

double calct(Point a, Point b, Point c, Vector u){
    Vector v = b-a;
    Point cp = crosspointLL(a,b,c,c+u);

    double t;
    if(EQ(v.X, 0)){
        if( EQ(v.Y, 0) ){
            while(1){}
        }

        t = (cp.Y - a.Y)/v.Y;
    }
    else{
        t = (cp.X - a.X)/v.X;
    }
    return t;
}

int main(){
    int n;
    cin >>n;

    VP p(n);
    rep(i,n) p[i] = READ();

    double ans = 1e12;
    int min_idx = 0, max_idx = 0;
    double mnt = INF, mxt = -INF;
    rep(i,n){
        Point a = p[0], b = p[1];
        Vector v = b-a;

        Vector mnv = v * polar((double)1.0, -2*PI/3);
        Vector mxv = v * polar((double)1.0, -PI/3);

        Point c = p[i];

        double tt = calct(a,b,c,mnv);
        if(tt < mnt){
            mnt = tt;
            min_idx = i;
        }

        tt = calct(a,b,c,mxv);
        if(tt > mxt){
            mxt = tt;
            max_idx = i;
        }
        ans = abs( (a+mnt*v) - (a+mxt*v) );
    }

    // dbg(ans);
    // printf(" 0 : min %d max %d\n", min_idx, max_idx);

    for(int i=1; i<n; ++i){
        Point a = p[i], b = p[(i+1)%n];
        Vector v = b-a;

        Vector mnv = v * polar((double)1.0, -2*PI/3);
        Vector mxv = v * polar((double)1.0, -PI/3);
        // dbg(v);
        // dbg(mnv);
        // dbg(mxv);

        rep(loop,n){
            double pt = calct(a,b,p[min_idx],mnv);
            double qt = calct(a,b,p[(min_idx+1)%n],mnv);

            if(qt > pt+EPS) break;
            min_idx = (min_idx+1)%n;
        }

        rep(loop,n){
            double pt = calct(a,b,p[max_idx],mxv);
            double qt = calct(a,b,p[(max_idx+1)%n],mxv);

            if(qt < pt-EPS) break;
            max_idx = (max_idx+1)%n;
        }

        // printf(" %d : min %d max %d\n", i,min_idx, max_idx);

        mnt = calct(a,b,p[min_idx],mnv);
        mxt = calct(a,b,p[max_idx],mxv);
        double dd = abs( (a+mnt*v) - (a+mxt*v) );
        ans = min(ans, dd);
    }

    printf("%.10Lf\n", ans);
    return 0;
}


### Prompt

Construct a Cpp code solution to the problem outlined:

Problem

Given a convex polygon consisting of $ N $ vertices.

When considering an equilateral triangle that includes all the vertices of the convex polygon, find the minimum value of the length of one side of the equilateral triangle.

Constraints

The input satisfies the following conditions.

* $ 3 \ le N \ le 10000 $
* $ -10 ^ 9 \ le px_i, py_i \ le 10 ^ 9 $
* No matter which of the three vertices of the convex polygon is selected, they do not exist on the same straight line.

Input

All inputs are given as integers in the following format:


$ N $
$ px_1 $ $ py_1 $
$ px_2 $ $ py_2 $
$ \ vdots $
$ px_N $ $ py_N $


The $ 1 $ line is given an integer $ N $ that represents the number of vertices of the convex polygon.
In the following $ N $ line, the information of each vertex of the convex polygon is given in the counterclockwise order.
In the $ 1 + i $ line, $ px_i $ and $ py_i $ representing the coordinates of the $ i $ th vertex are given separated by blanks.

Output

Output the minimum value of the length of one side of an equilateral triangle that satisfies the condition.
However, absolute or relative errors up to $ 10 ^ {-5} $ are acceptable.

Examples

Input

4
2 3
1 2
2 1
3 2


Output

3.04720672


Input

7
-96469 25422
-55204 -45592
-29140 -72981
98837 -86795
92303 63297
19059 96012
-67980 70342


Output

310622.35426197

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
#define rep(i,n) for(int (i)=0;(i)<(int)(n);++(i))
#define all(x) (x).begin(),(x).end()
#define pb push_back
#define fi first
#define se second
#define dbg(x) cout<<#x" = "<<((x))<<endl
template<class T,class U> ostream& operator<<(ostream& o, const pair<T,U> &p){o<<"("<<p.fi<<","<<p.se<<")";return o;}
template<class T> ostream& operator<<(ostream& o, const vector<T> &v){o<<"[";for(T t:v){o<<t<<",";}o<<"]";return o;}

#define double long double

/* 基本要素 */
typedef complex<double> Point;
typedef pair<Point, Point> Line;
typedef vector<Point> VP;
const double PI = acos(-1);
const double EPS = 1e-6; // 許容誤差^2
const double INF = 1e9;
#define X real()
#define Y imag()
#define LE(n,m) ((n) - (m) < EPS)
#define GE(n,m) (EPS > (m) - (n))
#define EQ(n,m) (abs((n)-(m)) < EPS)

// 内積　dot(a,b) = |a||b|cosθ
double dot(Point a, Point b) {
  return a.X*b.X + a.Y*b.Y;
}

// 外積　cross(a,b) = |a||b|sinθ
double cross(Point a, Point b) {
  return a.X*b.Y - a.Y*b.X;
}

// 点の進行方向
// !!! 誤差に注意 !!! (掛け算したものとかなり小さいものを比べているので)
int ccw(Point a, Point b, Point c) {
  b -= a;  c -= a;
  if (cross(b,c) >  EPS) return +1;  // counter clockwise
  if (cross(b,c) < -EPS) return -1;  // clockwise
  if (dot(b,c)   < -EPS) return +2;  // c--a--b on line
  if (norm(b) < norm(c)) return -2;  // a--b--c on line or a==b
  return 0;                          // a--c--b on line or a==c or b==c
}

// 2直線の交点
Point crosspointLL(Point a1, Point a2, Point b1, Point b2) {
  double d1 = cross(b2-b1, b1-a1);
  double d2 = cross(b2-b1, a2-a1);
  if (EQ(d1, 0) && EQ(d2, 0)) return a1;  // same line
  assert(!EQ(d2, 0)); // 交点がない
  return a1 + d1/d2 * (a2-a1);
}

using Vector = Point;

Point READ(){
    double x,y;
    cin >>x >>y;
    return Point(x,y);
}

double calct(Point a, Point b, Point c, Vector u){
    Vector v = b-a;
    Point cp = crosspointLL(a,b,c,c+u);

    double t;
    if(EQ(v.X, 0)){
        if( EQ(v.Y, 0) ){
            while(1){}
        }

        t = (cp.Y - a.Y)/v.Y;
    }
    else{
        t = (cp.X - a.X)/v.X;
    }
    return t;
}

int main(){
    int n;
    cin >>n;

    VP p(n);
    rep(i,n) p[i] = READ();

    double ans = 1e12;
    int min_idx = 0, max_idx = 0;
    double mnt = INF, mxt = -INF;
    rep(i,n){
        Point a = p[0], b = p[1];
        Vector v = b-a;

        Vector mnv = v * polar((double)1.0, -2*PI/3);
        Vector mxv = v * polar((double)1.0, -PI/3);

        Point c = p[i];

        double tt = calct(a,b,c,mnv);
        if(tt < mnt){
            mnt = tt;
            min_idx = i;
        }

        tt = calct(a,b,c,mxv);
        if(tt > mxt){
            mxt = tt;
            max_idx = i;
        }
        ans = abs( (a+mnt*v) - (a+mxt*v) );
    }

    // dbg(ans);
    // printf(" 0 : min %d max %d\n", min_idx, max_idx);

    for(int i=1; i<n; ++i){
        Point a = p[i], b = p[(i+1)%n];
        Vector v = b-a;

        Vector mnv = v * polar((double)1.0, -2*PI/3);
        Vector mxv = v * polar((double)1.0, -PI/3);
        // dbg(v);
        // dbg(mnv);
        // dbg(mxv);

        rep(loop,n){
            double pt = calct(a,b,p[min_idx],mnv);
            double qt = calct(a,b,p[(min_idx+1)%n],mnv);

            if(qt > pt+EPS) break;
            min_idx = (min_idx+1)%n;
        }

        rep(loop,n){
            double pt = calct(a,b,p[max_idx],mxv);
            double qt = calct(a,b,p[(max_idx+1)%n],mxv);

            if(qt < pt-EPS) break;
            max_idx = (max_idx+1)%n;
        }

        // printf(" %d : min %d max %d\n", i,min_idx, max_idx);

        mnt = calct(a,b,p[min_idx],mnv);
        mxt = calct(a,b,p[max_idx],mxv);
        double dd = abs( (a+mnt*v) - (a+mxt*v) );
        ans = min(ans, dd);
    }

    printf("%.10Lf\n", ans);
    return 0;
}


```