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;

const double EPS = 1e-8, PI = acos(-1);

inline bool eq(double a, double b) { return fabs(b - a) < EPS; }

struct Point
{
  double x, y;

  Point() {}

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

  Point operator+(const Point &b) const { return Point(x + b.x, y + b.y); }

  Point operator-(const Point &b) const { return Point(x - b.x, y - b.y); }

  Point operator*(const double b) const { return Point(x * b, y * b); }

  Point operator*(const Point &b) const { return Point(x * b.x - y * b.y, x * b.y + y * b.x); }

  Point operator/(const double b) const { return Point(x / b, y / b); }

  bool operator<(const Point &b) const { return x != b.x ? x < b.x : y < b.y; }

  bool operator==(const Point &b) const { return eq(x, b.x) && eq(y, b.y); }

  double norm() { return x * x + y * y; }

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

  double abs() { return sqrt(norm()); }

  Point rotate(double theta) { return Point(cos(theta) * x - sin(theta) * y, sin(theta) * x + cos(theta) * y); }

  Point rotate90() { return Point(-y, x); }

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

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

double cross(const Point &a, const Point &b)
{
  return a.x * b.y - a.y * b.x;
}

double dot(const Point &a, const Point &b)
{
  return a.x * b.x + a.y * b.y;
}

struct Line
{
  Point a, b;

  Line() {};

  Line(Point a, Point b) : a(a), b(b) {};

  friend ostream &operator<<(ostream &os, Line &p) { return os << "(" << p.a.x << "," << p.a.y << ") to (" << p.b.x << "," << p.b.y << ")"; }

  friend istream &operator>>(istream &is, Line &a) { return is >> a.a.x >> a.a.y >> a.b.x >> a.b.y; }
};

struct Segment
{
  Point a, b;

  Segment() {};

  Segment(Point a, Point b) : a(a), b(b) {};

  friend ostream &operator<<(ostream &os, Segment &p) { return os << "(" << p.a.x << "," << p.a.y << ") to (" << p.b.x << "," << p.b.y << ")"; }

  friend istream &operator>>(istream &is, Segment &a) { return is >> a.a.x >> a.a.y >> a.b.x >> a.b.y; }
};

int ccw(const Point &a, Point b, Point c)
{
  b = b - a, c = c - a;
  if(cross(b, c) > EPS) return +1;
  if(cross(b, c) < -EPS) return -1;
  if(dot(b, c) < 0) return +2;
  if(b.norm() < c.norm()) return -2;
  return 0;
}

bool Intersect(const Segment &s, const Segment &t)
{
  return ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 && ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0;
}

bool Intersect(const Segment &s, const Point &p)
{
  return ccw(s.a, s.b, p) == 0;
}

Point Projection(const Segment &l, const Point &p)
{
  double t = dot(p - l.a, l.a - l.b) / (l.a - l.b).norm();
  return l.a + (l.a - l.b) * t;
}

double Distance(const Segment &s, const Point &p)
{
  Point r = Projection(s, p);
  if(Intersect(s, r)) return (r - p).abs();
  return min((s.a - p).abs(), (s.b - p).abs());
}

double Distance(const Segment &a, const Segment &b)
{
  if(Intersect(a, b)) return 0;
  return min(min(Distance(a, b.a), Distance(a, b.b)), min(Distance(b, a.a), Distance(b, a.b)));
}

struct Circle
{
  Point p;
  double r;

  Circle() {};

  Circle(Point p, double r) : p(p), r(r) {};
};

typedef vector< Segment > Segments;

template< typename T = int >
void Warshall(vector< vector< T > > &g)
{
  for(int k = 0; k < g.size(); k++) {
    for(int i = 0; i < g.size(); i++) {
      for(int j = 0; j < g.size(); j++) {
        g[i][j] = min(g[i][j], g[i][k] + g[k][j]);
      }
    }
  }
}

int main()
{
  int W, M;
  while(cin >> W >> M, W) {
    vector< Segments > polys(M + 2);
    for(int i = 0; i < M; i++) {
      int sz;
      cin >> sz;
      vector< Point > pts(sz);
      for(auto &p : pts) cin >> p;
      for(int j = 0; j < sz; j++) {
        polys[i].emplace_back(pts[j], pts[(j + 1) % sz]);
      }
    }
    int s = M, t = M + 1;
    polys[s].emplace_back(Point(0, -1e9), Point(0, 1e9));
    polys[t].emplace_back(Point(W, -1e9), Point(W, 1e9));
    M += 2;

    vector< vector< double > > mat(M, vector< double >(M, 1e9));
    for(int i = 0; i < M; i++) {
      mat[i][i] = 0.0;
      for(int j = 0; j < i; j++) {
        double dist = 1e9;
        for(auto &s1 : polys[i]) {
          for(auto &s2 : polys[j]) {
            dist = min(dist, Distance(s1, s2));
          }
        }
        mat[i][j] = mat[j][i] = dist;
      }
    }
    Warshall< double >(mat);
    cout << fixed << setprecision(15) << mat[s][t] << endl;
  }

}