Problem: You are an officer of the Department of Land and Transport in Oykot City. The department has a plan to build a subway network in the city central of Oykot.

In the plan, n subway lines are built, and each line has two or more stations. Because of technical problems, a rail track between two stations should be straight and should not have any slope. To make things worse, no rail track can contact with another rail track, even on a station. In other words, two subways on the same floor cannot have any intersection.

Your job is to calculate the least number of required floors in their plan.



Input

The input consists of multiple datasets. Each dataset is formatted as follows:


N
Line1
Line2
...
LineN


Here, N is a positive integer that indicates the number of subway lines to be built in the plan (N ≤ 22), and Linei is the description of the i-th subway line with the following format:


S
X1 Y1
X2 Y2
...
XS YS


S is a positive integer that indicates the number of stations in the line (S ≤ 30), and (Xi, Yi) indicates the coordinates of the i-th station of the line (-10000 ≤ Xi, Yi ≤ 10000). The rail tracks are going to be built between two consecutive stations in the description. No stations of the same line have the same coordinates.

The input is terminated by a dataset of N = 0, and it should not be processed.

Output

For each dataset, you should output the least number of required floors.

Example

Input

2
2
0 0
10 0
2
0 10
10 10
2
2
0 0
10 10
2
0 10
10 0
3
2
0 0
10 10
2
0 10
10 0
2
1 0
1 10
0


Output

1
2
3

Solution (in CPP):

Solution:
#include <iostream>
#include <vector>
#include <map>
#include <cstring>

using namespace std;

constexpr double EPS = (1e-10);

struct Point {
    double x, y;

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

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

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

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

double norm(const Point& p)
{
    return dot(p, p);
}

constexpr int COUNTER_CLOCKWISE = +1;
constexpr int CLOCKWISE         = -1;
constexpr int ONLINE_BACK       = +2;
constexpr int ONLINE_FRONT      = -2;
constexpr int ON_SEGMENT        = +0;
using Vector = Point;

int ccw(const Point& p0, const Point& p1, const Point& p2)
{
    Vector a = p1 - p0;
    Vector 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;                        
}

struct Segment {
    Point s, t;
    Segment() {}
    Segment(Point s, Point t) : s{s}, t{t} {}
};

bool intersect(const Segment& a, const Segment& b)
{
    Point s[2]{a.s, a.t}, t[2]{b.s, b.t};
    return (ccw(s[0], s[1], t[0]) * ccw(s[0], s[1], t[1]) <= 0 &&
            ccw(t[0], t[1], s[0]) * ccw(t[0], t[1], s[1]) <= 0);
}

bool intersect(const vector<Point>& a, const vector<Point>& b)
{
    int as = a.size(), bs = b.size();
    for (int i = 0; i < as - 1; i++) {
        for (int j = 0; j < bs - 1; j++) {
            Segment s1{a[i], a[i + 1]}, s2{b[j], b[j + 1]};
            if (intersect(s1, s2)) {
                return 1;
            }
        }
    }
    return 0;
}

constexpr int MAX = 55; 

bool con[MAX][MAX];
map<vector<int>, int> mem;

void init(int N)
{
    mem.clear();
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            con[i][j] = 1;
        }
    }
}

using ll = long long;

inline int mod_pow(ll x, int n, int mod)
{
    if (n == 0) return 1;
    int res = mod_pow(x * x % mod, n / 2, mod);
    if (n & 1) res = res * x % mod;
    return res;
}

int I[1 << 22], bn[1 << 22];

bool c(int n, int k)
{
    ll g = 0;
    constexpr int MOD = 10009;    
    for (int i = 0; i < (1 << n); i++) {
        if (bn[i] % 2 == 0) {
            g += mod_pow(I[i], k, MOD);
        } else {
            g -= mod_pow(I[i], k, MOD);
        }
    }
    return (g % MOD + MOD) % MOD != 0;
}

int paint_color_num(int n)
{
    vector<int> N(1 << n);
    for (int i = 0; i < n; i++) {
        int bit = (1 << i);
        for (int j = 0; j < n; j++) {
            if (i != j && con[i][j]) {
                bit |= (1 << j);
            }
        }
        N[i] = bit;
    }
        
    memset(I, 0, sizeof(I));
    I[0] = 1;
    for (int S = 1; S < (1 << n); S++) {
        bn[S] = __builtin_popcount(S);
        int v = 0;
        while (!(S >> v & 1)) v++;
        I[S] = I[S - (1 << v)] + I[S & ~N[v]];
    }
            
    int L = 0, R = n;
    while (R - L > 1) {
        int mid = (L + R) / 2;
        if (c(n, mid)) {
            R = mid;
        } else {
            L = mid;
        }           
    }
    return R;
}

int main()
{
    int N;
    while (cin >> N, N) {
        vector<vector<Point>> ps(N);
        for (int i = 0; i < N; i++) {
            int S;
            cin >> S;
            ps[i].resize(S);
            for (int j = 0; j < S; j++) {
                cin >> ps[i][j].x >> ps[i][j].y;
            }
        }
        
        init(N);
                
        for (int i = 0; i < N; i++) {
            for (int j = i + 1; j < N; j++) {
                if (!intersect(ps[i], ps[j])) {
                    con[i][j] = con[j][i] = 0;
                }
            }
        }
        cout << paint_color_num(N) << endl;
    }        
    return 0;
}