Problem: You are given plans of rooms of polygonal shapes. The walls of the rooms on the plans are placed parallel to either x-axis or y-axis. In addition, the walls are made of special materials so they reflect light from sources as mirrors do, but only once. In other words, the walls do not reflect light already reflected at another point of the walls.

Now we have each room furnished with one lamp. Walls will be illuminated by the lamp directly or indirectly. However, since the walls reflect the light only once, some part of the walls may not be illuminated.

You are requested to write a program that calculates the total length of unilluminated part of the walls.

<image>

Figure 10: The room given as the second case in Sample Input



Input

The input consists of multiple test cases.

The first line of each case contains a single positive even integer N (4 ≤ N ≤ 20), which indicates the number of the corners. The following N lines describe the corners counterclockwise. The i-th line contains two integers xi and yi , where (xi , yi ) indicates the coordinates of the i-th corner. The last line of the case contains x' and y' , where (x' , y' ) indicates the coordinates of the lamp.

To make the problem simple, you may assume that the input meets the following conditions:

* All coordinate values are integers not greater than 100 in their absolute values.
* No two walls intersect or touch except for their ends.
* The walls do not intersect nor touch each other.
* The walls turn each corner by a right angle.
* The lamp exists strictly inside the room off the wall.
* The x-coordinate of the lamp does not coincide with that of any wall; neither does the y-coordinate.



The input is terminated by a line containing a single zero.

Output

For each case, output the length of the unilluminated part in one line. The output value may have an arbitrary number of decimal digits, but may not contain an error greater than 10-3 .

Example

Input

4
0 0
2 0
2 2
0 2
1 1
6
2 2
2 5
0 5
0 0
5 0
5 2
1 4
0


Output

0.000
3.000

Solution (in CPP):

Solution:
#include<bits/stdc++.h>

#define REP(i,s,n) for(int i=s;i<n;i++)
#define rep(i,n) REP(i,0,n)
#define EPS (1e-6)
#define COUNTER_CLOCKWISE 1
#define CLOCKWISE -1 
#define ONLINE_BACK 2
#define ONLINE_FRONT -2
#define ON_SEGMENT 0
#define equals(a,b) (fabs((a)-(b)) < EPS)

using namespace std;

class Point{
public:
  double x,y;

  Point(double x = 0,double y = 0): x(x),y(y){}

  Point operator + (Point p){return Point(x+p.x,y+p.y);}
  Point operator - (Point p){return Point(x-p.x,y-p.y);}
  Point operator * (double a){return Point(a*x,a*y);}
  Point operator / (double a){return Point(x/a,y/a);}
  Point operator * (Point a){ return Point(x * a.x - y * a.y, x * a.y + y * a.x); }
  bool operator < (const Point& p) const{ return !equals(x,p.x)?x<p.x:(!equals(y,p.y)&&y<p.y); }
  bool operator == (const Point& p)const{ return fabs(x-p.x) < EPS && fabs(y-p.y) < EPS; }
};

struct Segment{
  Point p1,p2;
  Segment(Point p1 = Point(),Point p2 = Point()):p1(p1),p2(p2){}
  bool operator <  (const Segment& s)const { return (p1==s.p1) ? p2 < s.p2 : p1 < s.p1; } 
  bool operator == (const Segment& p)const { return ( p.p1 == p1 && p.p2 == p2 ) || ( p.p1 == p2 && p.p2 == p1 ); }
};

typedef Point Vector;
typedef Segment Line;
typedef vector<Point> Polygon;

double dot(Point a,Point b){ return a.x*b.x + a.y*b.y; }
double cross(Point a,Point b){ return a.x*b.y - a.y*b.x; }
double norm(Point a){ return a.x*a.x+a.y*a.y; }
double abs(Point a){ return sqrt(norm(a)); }

int ccw(Point p0,Point p1,Point p2){
  Point a = p1-p0;
  Point b = p2-p0;
  if(cross(a,b) > EPS)return COUNTER_CLOCKWISE;
  if(cross(a,b) < -EPS)return CLOCKWISE;
  if(dot(a,b) < -EPS)return ONLINE_BACK;
  if(norm(a) < norm(b))return ONLINE_FRONT;
  return ON_SEGMENT;
}

bool intersectSS(Line s, Line t) {
  return ccw(s.p1,s.p2,t.p1)*ccw(s.p1,s.p2,t.p2) <= 0 &&
         ccw(t.p1,t.p2,s.p1)*ccw(t.p1,t.p2,s.p2) <= 0;
}

Point projection(Line l,Point p) {
  double t = dot(p-l.p1, l.p1-l.p2) / norm(l.p1-l.p2);
  return l.p1 + (l.p1-l.p2)*t;
}

Point reflection(Line l,Point p) { return p + (projection(l, p) - p) * 2; }

Point crosspoint(Line l,Line m){
  double A = cross(l.p2-l.p1,m.p2-m.p1);
  double B = cross(l.p2-l.p1,l.p2-m.p1);
  if(abs(A) < EPS && abs(B) < EPS){
    vector<Point> vec;
    vec.push_back(l.p1),vec.push_back(l.p2),vec.push_back(m.p1),vec.push_back(m.p2);
    sort(vec.begin(),vec.end());
    return vec[1];
  }
  if(abs(A) < EPS)assert(false);
  return m.p1 + (m.p2-m.p1)*(B/A);
}

double cross3p(Point p,Point q,Point r) { return (r.x-q.x) * (p.y -q.y) - (r.y - q.y) * (p.x - q.x); }
bool collinear(Point p,Point q,Point r) { return fabs(cross3p(p,q,r)) < EPS; }
 
bool onSegment(Point p,Point q,Point r){
  return collinear(p,q,r) && equals(sqrt(pow(p.x-r.x,2)+pow(p.y-r.y,2)) + sqrt(pow(r.x-q.x,2) + pow(r.y-q.y,2) ),sqrt(pow(p.x-q.x,2)+pow(p.y-q.y,2)) ) ;
}

double angle(Point a,Point b,Point c) {
  double ux = b.x - a.x, uy = b.y - a.y;
  double vx = c.x - a.x, vy = c.y - a.y;
  return acos((ux*vx + uy*vy)/sqrt((ux*ux + uy*uy) * (vx*vx + vy*vy)));
}  
  
//多角形poly内（線分上も含む）に点pが存在するかどうは判定する  
bool inPolygon(Polygon poly,Point p){
  if((int)poly.size() == 0)return false;
  rep(i,poly.size()) if(onSegment(poly[i],poly[(i+1)%poly.size()],p))return true;
  double sum = 0;
  for(int i=0; i < (int)poly.size() ;i++) {
    if( equals(cross(poly[i]-p,poly[(i+1)%poly.size()]-p),0.0) ) continue; // ３点が平行だとangleがnanを返しsumがnanになり死ぬ
    if( cross3p(p,poly[i],poly[(i+1)%poly.size()]) < 0 ) sum -= angle(p,poly[i],poly[(i+1)%poly.size()]);
    else                                                 sum += angle(p,poly[i],poly[(i+1)%poly.size()]);
  }
  // あまり誤差を厳しくしすぎると良くないので以下のほうが良い 
  const double eps = 1e-5;
  return (fabs(sum - 2*M_PI) < eps || fabs(sum + 2*M_PI) < eps);
}  

bool inPolygon(const Polygon &poly,Segment seg){
  vector<Point> vp;
  vp.push_back(seg.p1);
  vp.push_back(seg.p2);
  rep(i,(int)poly.size()) {
    Segment edge = Segment(poly[i],poly[(i+1)%(int)poly.size()]);
    if( equals(cross(seg.p1-seg.p2,edge.p1-edge.p2),0.0) ) {
      if( onSegment(seg.p1,seg.p2,edge.p1) ) vp.push_back(edge.p1);
      if( onSegment(seg.p1,seg.p2,edge.p2) ) vp.push_back(edge.p2);
    } else {
      if( intersectSS(seg,edge) ) vp.push_back(crosspoint(seg,edge));
    }
  }
  sort(vp.begin(),vp.end());
  vp.erase(unique(vp.begin(),vp.end()),vp.end());
  rep(i,(int)vp.size()-1) {
    Point middle_point = ( vp[i] + vp[i+1] ) / 2.0;
    if( !inPolygon(poly,middle_point) ) return false;
  }
  return true;  
}

int V;
Polygon poly;
Point light;

// type 0 => =w=, type 1 => mirror
void inner_add_new_points(Polygon &pol,vector<Point> &vp,int type,Line ref){
  rep(i,V){
    // ライトから普通のポリゴンの角をみる
    Vector e = ( poly[i] - light ) / abs( poly[i] - light );
    Segment seg = Segment(light,light+e*10000);
    rep(j,V) {
      Segment poly_seg = Segment(pol[j],pol[(j+1)%V]);
      if( equals(cross(seg.p2-seg.p1,poly_seg.p2-poly_seg.p1),0.0) ) continue;
      if( intersectSS(poly_seg,seg) ) {
        if( !type ) vp.push_back(crosspoint(poly_seg,seg));
        else        vp.push_back(reflection(ref,crosspoint(poly_seg,seg)));
      }
    }

    // ライトから反転したポリゴンの角をみる
    e = ( pol[i] - light ) / abs( pol[i] - light );
    seg = Segment(light,light+e*10000);
    rep(j,V) {
      Segment poly_seg = Segment(pol[j],pol[(j+1)%V]);
      if( equals(cross(seg.p2-seg.p1,poly_seg.p2-poly_seg.p1),0.0) ) continue;
      if( intersectSS(poly_seg,seg) ) {
        if( !type ) vp.push_back(crosspoint(poly_seg,seg));
        else        vp.push_back(reflection(ref,crosspoint(poly_seg,seg)));
      }
    }
  }
}

bool  LT(double a,double b) { return !equals(a,b) && a < b; } 
bool LTE(double a,double b) { return  equals(a,b) || a < b; } 

Point vir;
bool comp_dist(const Point& p1,const Point& p2){ return LT(abs(vir-p1),abs(vir-p2)); }

void add_new_points(Polygon &new_poly){
  vector<Point> vp;
  inner_add_new_points(poly,vp,0,Line());
  rep(i,V){
    Line ref = Line(poly[i],poly[(i+1)%V]);
    Polygon rpoly(V);
    rep(j,V) {
      rpoly[j] = reflection(ref,poly[j]);
      if( equals(rpoly[j].x,0.0) ) rpoly[j].x = 0;
      if( equals(rpoly[j].y,0.0) ) rpoly[j].y = 0;
    }
    inner_add_new_points(rpoly,vp,1,ref);
  }

  rep(i,(int)vp.size()) {
    if( equals(vp[i].x,0.0) ) vp[i].x = 0;
    if( equals(vp[i].y,0.0) ) vp[i].y = 0;
  }

  vector<Point> tmp;
  rep(i,vp.size()) {
    bool failed = false;
    rep(j,V) if( vp[i] == poly[j] ) { failed = true; break; }
    if( !failed ) tmp.push_back(vp[i]);
  }
  vp = tmp;

  sort(vp.begin(),vp.end());
  vp.erase(unique(vp.begin(),vp.end()),vp.end());

  rep(i,V){
    vector<Point> buf;
    Segment seg = Segment(poly[i],poly[(i+1)%V]);
    rep(j,(int)vp.size()) if( onSegment(seg.p1,seg.p2,vp[j]) ) buf.push_back(vp[j]);
    vir = seg.p1;
    sort(buf.begin(),buf.end(),comp_dist);
    rep(j,(int)buf.size()) new_poly.push_back(buf[j]);
    new_poly.push_back(seg.p2);
  }  
}

void compute_saw(Polygon &npoly,bool *saw){
  
  int nV = npoly.size();  
  // 直接見れる
  rep(i,nV) {
    Point mp = ( npoly[i] + npoly[(i+1)%nV] ) / 2.0;
    Segment vline = Segment(light,mp);
    bool failed = false;
    rep(j,V) {
      Segment seg = Segment(poly[j],poly[(j+1)%V]);
      if( equals(cross(seg.p1-seg.p2,vline.p1-vline.p2),0.0) ) continue;
      if( intersectSS(seg,vline) ) {
        if( onSegment(seg.p1,seg.p2,vline.p1) || onSegment(seg.p1,seg.p2,vline.p2) ) continue;
        if( onSegment(vline.p1,vline.p2,seg.p1) || onSegment(vline.p1,vline.p2,seg.p2) ) continue;
        failed = true;
        break;
      }
    }
    if( failed ) continue;
    saw[i] = true;
  }

  // 鏡一枚で見れる
  // 直接見える鏡だけを列挙
  vector<Line> ref;
  rep(i,nV) {
    Point mp = ( npoly[i] + npoly[(i+1)%nV] ) / 2.0;
    Segment vline = Segment(light,mp);
    bool failed = false;
    rep(j,V) {
      Segment seg = Segment(poly[j],poly[(j+1)%V]);
      if( equals(cross(seg.p1-seg.p2,vline.p1-vline.p2),0.0) ) continue;
      if( intersectSS(seg,vline) ) {
        if( onSegment(seg.p1,seg.p2,vline.p1) || onSegment(seg.p1,seg.p2,vline.p2) ) continue;
        if( onSegment(vline.p1,vline.p2,seg.p1) || onSegment(vline.p1,vline.p2,seg.p2) ) continue;
        failed = true;
        break;
      }
    }
    if( !failed ) ref.push_back(Line(npoly[i],npoly[(i+1)%nV]));
  }
  
  vector<Polygon> ref_polies;
  rep(i,(int)ref.size()){
    Polygon tmp_poly;
    rep(j,(int)poly.size()) tmp_poly.push_back(reflection(ref[i],poly[j]));
    ref_polies.push_back(tmp_poly);
  }

  rep(i,nV) if( !saw[i] ) {
    rep(j,(int)ref.size()){
      Segment rev_edge = Segment(reflection(ref[j],npoly[i]),reflection(ref[j],npoly[(i+1)%nV]));
      Point rev_mp = ( rev_edge.p1 + rev_edge.p2 ) / 2.0;
      Segment rev_vline = Segment(light,rev_mp);
      if( equals(cross(rev_vline.p1-rev_vline.p2,ref[j].p1-ref[j].p2),0.0) ) continue;
      if( !intersectSS(rev_vline,ref[j]) ) continue;
      rev_vline.p1 = crosspoint(rev_vline,ref[j]);
      if( inPolygon(ref_polies[j],rev_vline) ) {
        saw[i] = true;
        break;
      }
    }
  }
}

void compute(){
  Polygon npoly;
  add_new_points(npoly);
  int nV = npoly.size();
  bool saw[nV];
  rep(i,nV) saw[i] = false;
  compute_saw(npoly,saw);
  double sum = 0;
  rep(i,nV) if( !saw[i] )sum += abs(npoly[(i+1)%nV]-npoly[i]);
  sum = fabs(sum);
  printf("%.10lf\n",sum);
}

int main(){
  while( cin >> V, V ){
    poly.resize(V);
    rep(i,V) cin >> poly[i].x >> poly[i].y;
    cin >> light.x >> light.y;
    compute();
  }
  return 0;
}