Problem: Wind Corridor is a covered passageway where strong wind is always blowing. It is a long corridor of width W, and there are several pillars in it. Each pillar is a right prism and its face is a polygon (not necessarily convex).

In this problem, we consider two-dimensional space where the positive x-axis points the east and the positive y-axis points the north. The passageway spans from the south to the north, and its length is infinity. Specifically, it covers the area 0 ≤ x ≤ W. The outside of the passageway is filled with walls. Each pillar is expressed as a polygon, and all the pillars are located within the corridor without conflicting or touching each other.

Wind blows from the south side of the corridor to the north. For each second, w unit volume of air can be flowed at most if the minimum width of the path of the wind is w. Note that the path may fork and merge, but never overlaps with pillars and walls.

Your task in this problem is to write a program that calculates the maximum amount of air that can be flowed through the corridor per second.



Input

The input consists of multiple datasets. Each dataset has the following format:

The first line of the input contains two integers W and N. W is the width of the corridor, and N is the number of pillars. W and N satisfy the following condition: 1 ≤ W ≤ 104 and 0 ≤ N ≤ 200.

Then, N specifications of each pillar follow. Each specification starts with a line that contains a single integer M, which is the number of the vertices of a polygon (3 ≤ M ≤ 40). The following M lines describe the shape of the polygon. The i-th line (1 ≤ i ≤ M) contains two integers xi and yi that denote the coordinate of the i-th vertex (0 < xi < W, 0 < yi < 104).

The last dataset is followed by a line containing two zeros. This line is not a part of any dataset and should not be processed.

Output

For each dataset, your program should print a line that contains the maximum amount of air flow per second, in unit volume. The output may contain arbitrary number of digits after the decimal point, but the absolute error must not exceed 10-6.

Example

Input

5 2
4
1 1
1 2
2 2
2 1
4
3 3
3 4
4 4
4 3
0 0


Output

3.41421356

Solution (in CPP):

Solution:
#include<bits/stdc++.h>
using namespace std;
using Int = long long;
template<typename T1,typename T2> inline void chmin(T1 &a,T2 b){if(a>b) a=b;}
template<typename T1,typename T2> inline void chmax(T1 &a,T2 b){if(a<b) a=b;}



#define EPS (1e-10)
#define equals(a,b) (fabs((a)-(b)) < EPS)
#define PI 3.141592653589793238
 
// COUNTER CLOCKWISE
static const int CCW_COUNTER_CLOCKWISE = 1;
static const int CCW_CLOCKWISE = -1;
static const int CCW_ONLINE_BACK = 2;
static const int CCW_ONLINE_FRONT = -2;
static const int CCW_ON_SEGMENT = 0;

//Intercsect Circle & Circle
static const int ICC_SEPERATE = 4;
static const int ICC_CIRCUMSCRIBE = 3;
static const int ICC_INTERSECT = 2;
static const int ICC_INSCRIBE = 1;
static const int ICC_CONTAIN = 0;

struct Point{
  double x,y;
  Point(){}
  Point(double x,double y) :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 k){return Point(x*k,y*k);}
  Point operator/(double k){return Point(x/k,y/k);}
  double norm(){return x*x+y*y;}
  double abs(){return sqrt(norm());}

  bool operator < (const Point &p) const{
    return x!=p.x?x<p.x:y<p.y;
    //grid-point only
    //return !equals(x,p.x)?x<p.x:!equals(y,p.y)?y<p.y:0;
  }

  bool operator == (const Point &p) const{
    return fabs(x-p.x)<EPS && fabs(y-p.y)<EPS;
  }
};

struct EndPoint{
  Point p;
  int seg,st;
  EndPoint(){}
  EndPoint(Point p,int seg,int st):p(p),seg(seg),st(st){}
  bool operator<(const EndPoint &ep)const{
    if(p.y==ep.p.y) return st<ep.st;
    return p.y<ep.p.y;
  }
};

istream &operator >> (istream &is,Point &p){
  is>>p.x>>p.y;
  return is;
}

ostream &operator << (ostream &os,Point p){
  os<<fixed<<setprecision(12)<<p.x<<" "<<p.y;
  return os;
}

bool sort_x(Point a,Point b){
  return a.x!=b.x?a.x<b.x:a.y<b.y;
}

bool sort_y(Point a,Point b){
  return a.y!=b.y?a.y<b.y:a.x<b.x;
}

typedef Point Vector;
typedef vector<Point> Polygon;

istream &operator >> (istream &is,Polygon &p){
  for(int i=0;i<(int)p.size();i++) is>>p[i];
  return is;
}

struct Segment{
  Point p1,p2;
  Segment(){}
  Segment(Point p1, Point p2):p1(p1),p2(p2){}
};
typedef Segment Line;

istream &operator >> (istream &is,Segment &s){
  is>>s.p1>>s.p2;
  return is;
}

struct Circle{
  Point c;
  double r;
  Circle(){}
  Circle(Point c,double r):c(c),r(r){}
};

istream &operator >> (istream &is,Circle &c){
  is>>c.c>>c.r;
  return is;
}

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

Point orth(Point p){return Point(-p.y,p.x);}

bool isOrthogonal(Vector a,Vector b){
  return equals(dot(a,b),0.0);
}

bool isOrthogonal(Point a1,Point a2,Point b1,Point b2){
  return isOrthogonal(a1-a2,b1-b2);
}

bool isOrthogonal(Segment s1,Segment s2){
  return equals(dot(s1.p2-s1.p1,s2.p2-s2.p1),0.0);
}

bool isParallel(Vector a,Vector b){
  return equals(cross(a,b),0.0);
}

bool isParallel(Point a1,Point a2,Point b1,Point b2){
  return isParallel(a1-a2,b1-b2);
}

bool isParallel(Segment s1,Segment s2){
  return equals(cross(s1.p2-s1.p1,s2.p2-s2.p1),0.0); 
}

Point project(Segment s,Point p){
  Vector base=s.p2-s.p1;
  double r=dot(p-s.p1,base)/norm(base);
  return s.p1+base*r;
}

Point reflect(Segment s,Point p){
  return p+(project(s,p)-p)*2.0;
}

double arg(Vector p){
  return atan2(p.y,p.x);
}

Vector polar(double a,double r){
  return Point(cos(r)*a,sin(r)*a);
}

int ccw(Point p0,Point p1,Point p2);
bool intersectSS(Point p1,Point p2,Point p3,Point p4);
bool intersectSS(Segment s1,Segment s2);
bool intersectPS(Polygon p,Segment l);
int intersectCC(Circle c1,Circle c2);
bool intersectSC(Segment s,Circle c);
double getDistanceLP(Line l,Point p);
double getDistanceSP(Segment s,Point p);
double getDistanceSS(Segment s1,Segment s2);
Point getCrossPointSS(Segment s1,Segment s2);
Point getCrossPointLL(Line l1,Line l2);
Polygon getCrossPointCL(Circle c,Line l);
Polygon getCrossPointCC(Circle c1,Circle c2);
int contains(Polygon g,Point p);
Polygon andrewScan(Polygon s);
Polygon convex_hull(Polygon ps);
double diameter(Polygon s);
bool isConvex(Polygon p);
double area(Polygon s);
Polygon convexCut(Polygon p,Line l);
Line bisector(Point p1,Point p2);
Vector translate(Vector v,double theta);
vector<Line> corner(Line l1,Line l2);
vector<vector<pair<int, double> > >
segmentArrangement(vector<Segment> &ss, Polygon &ps);
  
int ccw(Point p0,Point p1,Point p2){
  Vector a = p1-p0;
  Vector b = p2-p0;
  if(cross(a,b) > EPS) return CCW_COUNTER_CLOCKWISE;
  if(cross(a,b) < -EPS) return CCW_CLOCKWISE;
  if(dot(a,b) < -EPS) return CCW_ONLINE_BACK;
  if(a.norm()<b.norm()) return CCW_ONLINE_FRONT;
  return CCW_ON_SEGMENT;
}

bool intersectSS(Point p1,Point p2,Point p3,Point p4){
  return (ccw(p1,p2,p3)*ccw(p1,p2,p4) <= 0 &&
          ccw(p3,p4,p1)*ccw(p3,p4,p2) <= 0 );
}

bool intersectSS(Segment s1,Segment s2){
  return intersectSS(s1.p1,s1.p2,s2.p1,s2.p2);
}

bool intersectPS(Polygon p,Segment l){
  int n=p.size();
  for(int i=0;i<n;i++)
    if(intersectSS(Segment(p[i],p[(i+1)%n]),l)) return 1;
  return 0;
}

int intersectCC(Circle c1,Circle c2){
  if(c1.r<c2.r) swap(c1,c2);
  double d=abs(c1.c-c2.c);
  double r=c1.r+c2.r;
  if(equals(d,r)) return ICC_CIRCUMSCRIBE;
  if(d>r) return ICC_SEPERATE;
  if(equals(d+c2.r,c1.r)) return ICC_INSCRIBE;
  if(d+c2.r<c1.r) return ICC_CONTAIN;
  return ICC_INTERSECT;
}

bool intersectSC(Segment s,Circle c){
  return getDistanceSP(s,c.c)<=c.r;
}

int intersectCS(Circle c,Segment s){
  if(norm(project(s,c.c)-c.c)-c.r*c.r>EPS) return 0;
  double d1=abs(c.c-s.p1),d2=abs(c.c-s.p2);
  if(d1<c.r+EPS&&d2<c.r+EPS) return 0;
  if((d1<c.r-EPS&&d2>c.r+EPS)||(d1>c.r+EPS&&d2<c.r-EPS)) return 1;
  Point h=project(s,c.c);
  if(dot(s.p1-h,s.p2-h)<0) return 2;
  return 0;
}

double getDistanceLP(Line l,Point p){
  return abs(cross(l.p2-l.p1,p-l.p1)/abs(l.p2-l.p1));
}

double getDistanceSP(Segment s,Point p){
  if(dot(s.p2-s.p1,p-s.p1) < 0.0 ) return abs(p-s.p1);
  if(dot(s.p1-s.p2,p-s.p2) < 0.0 ) return abs(p-s.p2);
  return getDistanceLP(s,p);
}

double getDistanceSS(Segment s1,Segment s2){
  if(intersectSS(s1,s2)) return 0.0;
  return min(min(getDistanceSP(s1,s2.p1),getDistanceSP(s1,s2.p2)),
             min(getDistanceSP(s2,s1.p1),getDistanceSP(s2,s1.p2)));
}

Point getCrossPointSS(Segment s1,Segment s2){
  for(int k=0;k<2;k++){
    if(getDistanceSP(s1,s2.p1)<EPS) return s2.p1; 
    if(getDistanceSP(s1,s2.p2)<EPS) return s2.p2;
    swap(s1,s2);
  }
  Vector base=s2.p2-s2.p1;
  double d1=abs(cross(base,s1.p1-s2.p1));
  double d2=abs(cross(base,s1.p2-s2.p1));
  double t=d1/(d1+d2);
  return s1.p1+(s1.p2-s1.p1)*t;
}

Point getCrossPointLL(Line l1,Line l2){
  double a=cross(l1.p2-l1.p1,l2.p2-l2.p1);
  double b=cross(l1.p2-l1.p1,l1.p2-l2.p1);
  if(abs(a)<EPS&&abs(b)<EPS) return l2.p1;
  return l2.p1+(l2.p2-l2.p1)*(b/a);
}

Polygon getCrossPointCL(Circle c,Line l){
  Polygon ps;
  Point pr=project(l,c.c);
  Vector e=(l.p2-l.p1)/abs(l.p2-l.p1);
  if(equals(getDistanceLP(l,c.c),c.r)){
    ps.emplace_back(pr);
    return ps;
  }
  double base=sqrt(c.r*c.r-norm(pr-c.c));
  ps.emplace_back(pr+e*base);
  ps.emplace_back(pr-e*base);
  return ps;
}

Polygon getCrossPointCS(Circle c,Segment s){
  Line l(s);
  Polygon res=getCrossPointCL(c,l);
  if(intersectCS(c,s)==2) return res;
  if(res.size()>1u){
    if(dot(l.p1-res[0],l.p2-res[0])>0) swap(res[0],res[1]);
    res.pop_back();
  }
  return res;
}


Polygon getCrossPointCC(Circle c1,Circle c2){
  Polygon p(2);
  double d=abs(c1.c-c2.c);
  double a=acos((c1.r*c1.r+d*d-c2.r*c2.r)/(2*c1.r*d));
  double t=arg(c2.c-c1.c);
  p[0]=c1.c+polar(c1.r,t+a);
  p[1]=c1.c+polar(c1.r,t-a);
  return p;
}

// IN:2 ON:1 OUT:0
int contains(Polygon g,Point p){
  int n=g.size();
  bool x=false;
  for(int i=0;i<n;i++){
    Point a=g[i]-p,b=g[(i+1)%n]-p;
    if(fabs(cross(a,b)) < EPS && dot(a,b) < EPS) return 1;
    if(a.y>b.y) swap(a,b);
    if(a.y < EPS && EPS < b.y && cross(a,b) > EPS ) x = !x;
  }
  return (x?2:0);
}

Polygon andrewScan(Polygon s){
  Polygon u,l;
  if(s.size()<3) return s;
  sort(s.begin(),s.end());
  u.push_back(s[0]);
  u.push_back(s[1]);
  l.push_back(s[s.size()-1]);
  l.push_back(s[s.size()-2]);
  for(int i=2;i<(int)s.size();i++){
    for(int n=u.size();n>=2&&ccw(u[n-2],u[n-1],s[i])!=CCW_CLOCKWISE;n--){
      u.pop_back();
    }
    u.push_back(s[i]);
  } 
  for(int i=s.size()-3;i>=0;i--){
    for(int n=l.size();n>=2&&ccw(l[n-2],l[n-1],s[i])!=CCW_CLOCKWISE;n--){
      l.pop_back();
    }
    l.push_back(s[i]);
  }
  reverse(l.begin(),l.end());
  for(int i=u.size()-2;i>=1;i--) l.push_back(u[i]);
  return l;
} 

Polygon convex_hull(Polygon ps){
  int n=ps.size();
  sort(ps.begin(),ps.end(),sort_y);
  int k=0;
  Polygon qs(n*2);
  for(int i=0;i<n;i++){
    while(k>1&&cross(qs[k-1]-qs[k-2],ps[i]-qs[k-1])<0) k--;
    qs[k++]=ps[i];
  }
  for(int i=n-2,t=k;i>=0;i--){
    while(k>t&&cross(qs[k-1]-qs[k-2],ps[i]-qs[k-1])<0) k--;
    qs[k++]=ps[i];
  }
  qs.resize(k-1);
  return qs;
}

double diameter(Polygon s){
  Polygon p=s;
  int n=p.size();
  if(n==2) return abs(p[0]-p[1]);
  int i=0,j=0;
  for(int k=0;k<n;k++){
    if(p[i]<p[k]) i=k;
    if(!(p[j]<p[k])) j=k;
  }
  double res=0;
  int si=i,sj=j;
  while(i!=sj||j!=si){
    res=max(res,abs(p[i]-p[j]));
    if(cross(p[(i+1)%n]-p[i],p[(j+1)%n]-p[j])<0.0){
      i=(i+1)%n;
    }else{
      j=(j+1)%n;
    }
  }
  return res;
}

bool isConvex(Polygon p){
  bool f=1;
  int n=p.size();
  for(int i=0;i<n;i++){
    int t=ccw(p[(i+n-1)%n],p[i],p[(i+1)%n]);
    f&=t!=CCW_CLOCKWISE;
  }
  return f;
}

double area(Polygon s){
  double res=0;
  for(int i=0;i<(int)s.size();i++){
    res+=cross(s[i],s[(i+1)%s.size()])/2.0;
  }
  return abs(res);
}

double area(Circle c1,Circle c2){
  double d=abs(c1.c-c2.c);
  if(c1.r+c2.r<=d+EPS) return 0;
  if(d<=abs(c1.r-c2.r)){
    double r=min(c1.r,c2.r);
    return PI*r*r;
  }
  double rc=(d*d+c1.r*c1.r-c2.r*c2.r)/(2*d);
  double th=acos(rc/c1.r);
  double ph=acos((d-rc)/c2.r);
  return c1.r*c1.r*th+c2.r*c2.r*ph-d*c1.r*sin(th);
}

Polygon convexCut(Polygon p,Line l){
  Polygon q;
  for(int i=0;i<(int)p.size();i++){
    Point a=p[i],b=p[(i+1)%p.size()];
    if(ccw(l.p1,l.p2,a)!=-1) q.push_back(a);
    if(ccw(l.p1,l.p2,a)*ccw(l.p1,l.p2,b)<0)
      q.push_back(getCrossPointLL(Line(a,b),l));
  }
  return q;
}

Line bisector(Point p1,Point p2){
  Circle c1=Circle(p1,abs(p1-p2)),c2=Circle(p2,abs(p1-p2));
  Polygon p=getCrossPointCC(c1,c2);
  if(cross(p2-p1,p[0]-p1)>0) swap(p[0],p[1]);
  return Line(p[0],p[1]);
}

Vector translate(Vector v,double theta){
  Vector res;
  res.x=cos(theta)*v.x-sin(theta)*v.y;
  res.y=sin(theta)*v.x+cos(theta)*v.y;
  return res;
}

vector<Line> corner(Line l1,Line l2){
  vector<Line> res;
  if(isParallel(l1,l2)){
    double d=getDistanceLP(l1,l2.p1)/2.0;
    Vector v1=l1.p2-l1.p1;
    v1=v1/v1.abs()*d;
    Point p=l2.p1+translate(v1,90.0*(PI/180.0));
    double d1=getDistanceLP(l1,p);
    double d2=getDistanceLP(l2,p);
    if(abs(d1-d2)>d){
      p=l2.p1+translate(v1,-90.0*(PI/180.0));
    }
    res.push_back(Line(p,p+v1));
  }else{
    Point p=getCrossPointLL(l1,l2);
    Vector v1=l1.p2-l1.p1,v2=l2.p2-l2.p1;
    v1=v1/v1.abs();
    v2=v2/v2.abs();
    res.push_back(Line(p,p+(v1+v2)));
    res.push_back(Line(p,p+translate(v1+v2,90.0*(PI/180.0))));
  }
  return res;
}

Polygon tangent(Circle c1,Point p2){
  Circle c2=Circle(p2,sqrt(norm(c1.c-p2)-c1.r*c1.r));
  Polygon p=getCrossPointCC(c1,c2);
  sort(p.begin(),p.end());
  return p;
}

vector<Line> tangent(Circle c1,Circle c2){
  vector<Line> ls;
  if(c1.r<c2.r) swap(c1,c2);
  double g=norm(c1.c-c2.c);
  if(equals(g,0)) return ls;
  Point u=(c2.c-c1.c)/sqrt(g);
  Point v=orth(u);
  for(int s=1;s>=-1;s-=2){
    double h=(c1.r+s*c2.r)/sqrt(g);
    if(equals(1-h*h,0)){
      ls.emplace_back(c1.c+u*c1.r,c1.c+(u+v)*c1.r);
    }else if(1-h*h>0){
      Point uu=u*h,vv=v*sqrt(1-h*h);
      ls.emplace_back(c1.c+(uu+vv)*c1.r,c2.c-(uu+vv)*c2.r*s);
      ls.emplace_back(c1.c+(uu-vv)*c1.r,c2.c-(uu-vv)*c2.r*s);
    }
  }
  
  return ls;
}

double closest_pair(Polygon &a,int l=0,int r=-1){
  if(r<0){
    r=a.size();
    sort(a.begin(),a.end(),sort_x);
  }
  if(r-l<=1) return abs(a[0]-a[1]);
  int m=(l+r)>>1;
  double x=a[m].x;
  double d=min(closest_pair(a,l,m),closest_pair(a,m,r));
  inplace_merge(a.begin()+l,a.begin()+m,a.begin()+r,sort_y);

  Polygon b;
  for(int i=l;i<r;i++){
    if(fabs(a[i].x-x)>=d) continue;
    for(int j=0;j<(int)b.size();j++){
      double dy=a[i].y-next(b.rbegin(),j)->y;
      if(dy>=d) break;
      d=min(d,abs(a[i]-*next(b.rbegin(),j)));
    }
    b.emplace_back(a[i]);
  }
  return d;
}

vector<vector<pair<int, double> > >
segmentArrangement(vector<Segment> &ss, Polygon &ps){
  int n=ss.size();
  for(int i=0;i<n;i++){
    ps.emplace_back(ss[i].p1);
    ps.emplace_back(ss[i].p2);
    for(int j=i+1;j<n;j++)
      if(intersectSS(ss[i],ss[j]))
        ps.emplace_back(getCrossPointSS(ss[i],ss[j]));
  }
  sort(ps.begin(),ps.end());
  ps.erase(unique(ps.begin(),ps.end()),ps.end());

  vector<vector<pair<int, double> > > G(ps.size());
  for(int i=0;i<n;i++){
    vector<pair<double,int> > ls;
    for(int j=0;j<(int)ps.size();j++)
      if(getDistanceSP(ss[i],ps[j])<EPS)
        ls.emplace_back(make_pair(norm(ss[i].p1-ps[j]),j));
      
    sort(ls.begin(),ls.end());
    for(int j=0;j+1<(int)ls.size();j++){
      int a=ls[j].second,b=ls[j+1].second;
      G[a].emplace_back(b,abs(ps[a]-ps[b]));
      G[b].emplace_back(a,abs(ps[a]-ps[b]));
    }
  }
  return G;
}

int manhattanIntersection(vector<Segment> ss,const int INF){
  const int BTM = 0;
  const int LFT = 1;
  const int RGH = 2;
  const int TOP = 3;
  
  int n=ss.size();
  vector<EndPoint> ep;
  for(int i=0;i<n;i++){
    if(ss[i].p1.y==ss[i].p2.y){
      if(ss[i].p1.x>ss[i].p2.x) swap(ss[i].p1,ss[i].p2);
      ep.emplace_back(ss[i].p1,i,LFT);
      ep.emplace_back(ss[i].p2,i,RGH);
    }else{
      if(ss[i].p1.y>ss[i].p2.y) swap(ss[i].p1,ss[i].p2);      
      ep.emplace_back(ss[i].p1,i,BTM);
      ep.emplace_back(ss[i].p2,i,TOP);
    }    
  }   
  sort(ep.begin(),ep.end());

  set<int> bt;
  bt.insert(INF);
  
  int cnt=0;
  for(int i=0;i<n*2;i++){
    if(ep[i].st==TOP){
      bt.erase(ep[i].p.x);
    }else if(ep[i].st==BTM){
      bt.emplace(ep[i].p.x);
    }else if(ep[i].st==LFT){
      auto b=bt.lower_bound(ss[ep[i].seg].p1.x);
      auto e=bt.upper_bound(ss[ep[i].seg].p2.x);
      cnt+=distance(b,e);
    }    
  }
  
  return cnt;
}

double area(Polygon ps,Circle c){
  if(ps.size()<3u) return 0;
  function<double(Circle, Point, Point)> dfs=
    [&](Circle c,Point a,Point b){
      Vector va=c.c-a,vb=c.c-b;
      double f=cross(va,vb),res=0;
      if(equals(f,0.0)) return res;
      if(max(abs(va),abs(vb))<c.r+EPS) return f;
      Vector d(dot(va,vb),cross(va,vb));
      if(getDistanceSP(Segment(a,b),c.c)>c.r-EPS)
        return c.r*c.r*atan2(d.y,d.x);
      auto u=getCrossPointCS(c,Segment(a,b));
      if(u.empty()) return res;
      if(u.size()>1u&&dot(u[1]-u[0],a-u[0])>0) swap(u[0],u[1]);
      u.emplace(u.begin(),a);
      u.emplace_back(b);
      for(int i=1;i<(int)u.size();i++)
        res+=dfs(c,u[i-1],u[i]);
      return res;
    };
  double res=0;
  for(int i=0;i<(int)ps.size();i++)
    res+=dfs(c,ps[i],ps[(i+1)%ps.size()]);
  return res/2;
}



template <typename T>
vector<T> dijkstra(int s,vector<vector<pair<int, T> > > & G,T INF){
  using P = pair<T, int>;
  int n=G.size();
  vector<T> d(n,INF);
  vector<int> b(n,-1);
  priority_queue<P,vector<P>,greater<P> > q;
  d[s]=0;
  q.emplace(d[s],s);
  while(!q.empty()){
    P p=q.top();q.pop();
    int v=p.second;
    if(d[v]<p.first) continue;
    for(auto& e:G[v]){
      int u=e.first;
      T c=e.second;
      if(d[u]>d[v]+c){
        d[u]=d[v]+c;
        b[u]=v;
        q.emplace(d[u],u);
      }
    }
  }
  return d;
}


struct Precision{
  Precision(){
    cout<<fixed<<setprecision(12);
  }
}precision_beet;

//INSERT ABOVE HERE

signed main(){
  int w,n;
  while(cin>>w>>n,w||n){
    vector<Polygon> ps(n);
    for(int i=0;i<n;i++){
      int m;
      cin>>m;
      ps[i].resize(m);
      cin>>ps[i];
    }
  
    using P = pair<int, double>;
    const double INF = 1e15;
    vector<vector<P > > G(n+2);
    
    for(int i=0;i<n;i++){
      double res=INF;
      Line l(Point(0,0),Point(0,1000));
      for(auto p:ps[i]) chmin(res,getDistanceLP(l,p));    
      G[n].emplace_back(i,res);    
    }
    for(int i=0;i<n;i++){
      double res=INF;
      Line l(Point(w,0),Point(w,1000));
      for(auto p:ps[i]) chmin(res,getDistanceLP(l,p));    
      G[i].emplace_back(n+1,res);    
    }
    G[n].emplace_back(n+1,w);
  
    for(int i=0;i<n;i++){
      for(int j=0;j<i;j++){
        double res=INF;
        for(int a=0;a<(int)ps[i].size();a++){          
          Segment la(ps[i][a],ps[i][(a+1)%ps[i].size()]);
          for(int b=0;b<(int)ps[j].size();b++){
            Segment lb(ps[j][b],ps[j][(b+1)%ps[j].size()]);
            chmin(res,getDistanceSP(la,ps[j][b]));
            chmin(res,getDistanceSP(lb,ps[i][a]));
          }
        }
        G[i].emplace_back(j,res);
        G[j].emplace_back(i,res);
      }    
    }  
    cout<<dijkstra(n,G,INF)[n+1]<<endl;
  }
  return 0;
}