#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-8)
#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;
 
const int IINF = INT_MAX;

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);}
  bool operator < (const Point& p) const{ return !equals(x,p.x)?x<p.x: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& 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 intersectLS(Line l, Line s) {
  return cross(l.p2-l.p1, s.p1-l.p1)*       // s[0] is left of l
         cross(l.p2-l.p1, s.p2-l.p1) < EPS; // s[1] is right of l
}

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 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());
    assert(vec[1] == vec[2]);
    return vec[1];
  }
  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 ccwtest(Point p,Point q,Point r){ return cross3p(p,q,r) > 0; }
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)));
}  
   
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()]);
  }
  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-edge.p1,seg.p2-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;
}
 
const int MAX_N = 110;
int N;
Point ps[MAX_N];
 
bool LTE(double a,double b) { return equals(a,b) || a < b ; } 
 
void compute(){
  vector<Point> poly;
  rep(i,N) poly.push_back(ps[i]);
  vector<Segment> segs;
  rep(i,N) segs.push_back(Segment(ps[i],ps[(i+1)%N]));
 
  double ans = 0;
  rep(i,N) REP(j,i+1,N) {
    Line line = Line(ps[i],ps[j]);
    vector<Point> cand;
    cand.push_back(line.p1),cand.push_back(line.p2);
    rep(k,N) {
      Segment seg = Segment(ps[k],ps[(k+1)%N]);
      if( !( line == seg ) && intersectLS(line,seg) && !equals(cross(line.p1-line.p2,seg.p1-seg.p2),0.0) ) cand.push_back(crosspoint(line,seg));
    }
    sort(cand.begin(),cand.end());
    cand.erase(unique(cand.begin(),cand.end()),cand.end());
    line.p1 = ps[i], line.p2 = ps[j];
    rep(k,(int)cand.size()) if( cand[k] == line.p1 ) {
      int pos = k;
      while( pos-1 >= 0 && inPolygon(poly,(line.p1+cand[pos-1])/2.0) ) line.p1 = cand[--pos];
      break;
    }
    rep(k,(int)cand.size()) if( cand[k] == line.p2 ) {
      int pos = k;
      while( pos+1 < cand.size() && inPolygon(poly,(line.p2+cand[pos+1])/2.0) ) line.p2 = cand[++pos];
      break;
    }

    bool failed = false;
    rep(k,N) {
      if( onSegment(line.p1,line.p2,segs[k].p1) || onSegment(line.p1,line.p2,segs[k].p2) ) continue;
      if( intersectSS(line,segs[k]) ) { 
        Point cp = crosspoint(line,segs[k]);
        if( cp == line.p1 || cp == line.p2 ) continue;
        failed = true; break; }
    }
    if( failed ) continue;
    if( !inPolygon(poly,line) ) continue;
    ans = max(ans,abs(line.p1-line.p2));
  }
  printf("%.8lf\n",ans);
}
 
int main(){
  int CNT = 1;
  while( scanf("%d",&N), N ){
    rep(i,N) scanf("%lf %lf",&ps[i].x,&ps[i].y);
    cout << "Case " << CNT++ << ": ";
    compute();
  }
  return 0;
}
### Prompt

Develop a solution in CPP to the problem described below:

Mr. KM, the mayor of KM city, decided to build a new elementary school. The site for the school has an awkward polygonal shape, which caused several problems. The most serious problem was that there was not enough space for a short distance racetrack. Your task is to help Mr. KM to calculate the maximum possible length for the racetrack that can be built in the site. The track can be considered as a straight line segment whose width can be ignored. The boundary of the site has a simple polygonal shape without self-intersection, and the track can touch the boundary. Note that the boundary might not be convex.



Input

The input consists of multiple test cases, followed by a line containing "0". Each test case has the following format. The first line contains an integer N (3 \leq N \leq 100). Each of the following N lines contains two integers x_i and y_i (-1,000 \leq x_i, y_i \leq 1,000), which describe the coordinates of a vertex of the polygonal border of the site, in counterclockwise order.

Output

For each test case, print its case number and the maximum possible length of the track in a line. The answer should be given as a floating point number with an absolute error of at most 10^{-6}.

Example

Input

4
0 0
10 0
10 10
0 10
3
0 0
1 0
0 1
0


Output

Case 1: 14.142135624
Case 2: 1.41421356

### Response

```cpp
#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-8)
#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;
 
const int IINF = INT_MAX;

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);}
  bool operator < (const Point& p) const{ return !equals(x,p.x)?x<p.x: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& 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 intersectLS(Line l, Line s) {
  return cross(l.p2-l.p1, s.p1-l.p1)*       // s[0] is left of l
         cross(l.p2-l.p1, s.p2-l.p1) < EPS; // s[1] is right of l
}

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 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());
    assert(vec[1] == vec[2]);
    return vec[1];
  }
  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 ccwtest(Point p,Point q,Point r){ return cross3p(p,q,r) > 0; }
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)));
}  
   
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()]);
  }
  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-edge.p1,seg.p2-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;
}
 
const int MAX_N = 110;
int N;
Point ps[MAX_N];
 
bool LTE(double a,double b) { return equals(a,b) || a < b ; } 
 
void compute(){
  vector<Point> poly;
  rep(i,N) poly.push_back(ps[i]);
  vector<Segment> segs;
  rep(i,N) segs.push_back(Segment(ps[i],ps[(i+1)%N]));
 
  double ans = 0;
  rep(i,N) REP(j,i+1,N) {
    Line line = Line(ps[i],ps[j]);
    vector<Point> cand;
    cand.push_back(line.p1),cand.push_back(line.p2);
    rep(k,N) {
      Segment seg = Segment(ps[k],ps[(k+1)%N]);
      if( !( line == seg ) && intersectLS(line,seg) && !equals(cross(line.p1-line.p2,seg.p1-seg.p2),0.0) ) cand.push_back(crosspoint(line,seg));
    }
    sort(cand.begin(),cand.end());
    cand.erase(unique(cand.begin(),cand.end()),cand.end());
    line.p1 = ps[i], line.p2 = ps[j];
    rep(k,(int)cand.size()) if( cand[k] == line.p1 ) {
      int pos = k;
      while( pos-1 >= 0 && inPolygon(poly,(line.p1+cand[pos-1])/2.0) ) line.p1 = cand[--pos];
      break;
    }
    rep(k,(int)cand.size()) if( cand[k] == line.p2 ) {
      int pos = k;
      while( pos+1 < cand.size() && inPolygon(poly,(line.p2+cand[pos+1])/2.0) ) line.p2 = cand[++pos];
      break;
    }

    bool failed = false;
    rep(k,N) {
      if( onSegment(line.p1,line.p2,segs[k].p1) || onSegment(line.p1,line.p2,segs[k].p2) ) continue;
      if( intersectSS(line,segs[k]) ) { 
        Point cp = crosspoint(line,segs[k]);
        if( cp == line.p1 || cp == line.p2 ) continue;
        failed = true; break; }
    }
    if( failed ) continue;
    if( !inPolygon(poly,line) ) continue;
    ans = max(ans,abs(line.p1-line.p2));
  }
  printf("%.8lf\n",ans);
}
 
int main(){
  int CNT = 1;
  while( scanf("%d",&N), N ){
    rep(i,N) scanf("%lf %lf",&ps[i].x,&ps[i].y);
    cout << "Case " << CNT++ << ": ";
    compute();
  }
  return 0;
}
```