#include <functional>
#include <algorithm>
#include <iostream>
#include <iomanip>
#include <numeric>
#include <sstream>
#include <cstring>
#include <cassert>
#include <cstdio>
#include <string>
#include <vector>
#include <bitset>
#include <queue>
#include <stack>
#include <cmath>
#include <ctime>
#include <list>
#include <set>
#include <map>
using namespace std;
#define REP(i, n) for (int i=0;i<int(n);++i)
#define FOR(i, a, b) for (int i=int(a);i<int(b);++i)
#define DWN(i, b, a) for (int i=int(b-1);i>=int(a);--i)
#define REP_1(i, n) for (int i=1;i<=int(n);++i)
#define FOR_1(i, a, b) for (int i=int(a);i<=int(b);++i)
#define DWN_1(i, b, a) for (int i=int(b);i>=int(a);--i)
#define REP_C(i, n) for (int n____=int(n),i=0;i<n____;++i)
#define FOR_C(i, a, b) for (int b____=int(b),i=a;i<b____;++i)
#define DWN_C(i, b, a) for (int a____=int(a),i=b-1;i>=a____;--i)
#define REP_N(i, n) for (i=0;i<int(n);++i)
#define FOR_N(i, a, b) for (i=int(a);i<int(b);++i)
#define DWN_N(i, b, a) for (i=int(b-1);i>=int(a);--i)
#define REP_1_C(i, n) for (int n____=int(n),i=1;i<=n____;++i)
#define FOR_1_C(i, a, b) for (int b____=int(b),i=a;i<=b____;++i)
#define DWN_1_C(i, b, a) for (int a____=int(a),i=b;i>=a____;--i)
#define REP_1_N(i, n) for (i=1;i<=int(n);++i)
#define FOR_1_N(i, a, b) for (i=int(a);i<=int(b);++i)
#define DWN_1_N(i, b, a) for (i=int(b);i>=int(a);--i)
#define REP_C_N(i, n) for (n____=int(n),i=0;i<n____;++i)
#define FOR_C_N(i, a, b) for (b____=int(b),i=a;i<b____;++i)
#define DWN_C_N(i, b, a) for (a____=int(a),i=b-1;i>=a____;--i)
#define REP_1_C_N(i, n) for (n____=int(n),i=1;i<=n____;++i)
#define FOR_1_C_N(i, a, b) for (b____=int(b),i=a;i<=b____;++i)
#define DWN_1_C_N(i, b, a) for (a____=int(a),i=b;i>=a____;--i)
//#define ECH(it, A) for (typeof(A.begin()) it=A.begin(); it != A.end(); ++it)
#define ECH(it, A) for (__typeof(A.begin()) it=A.begin(); it != A.end(); ++it)
#define DO(n) for ( int ____n ## __line__ = n; ____n ## __line__ -- ; )
#define _TO(i, a, b) int s_=a<b?1:-1,b_=b+s_;for(int i=a+s_;i!=b_;i+=s_)
#define TO(i, a, b) int s_=a<b?1:-1,b_=b+s_;for(int i=a;i!=b_;i+=s_)
#define TO_1(i, a, b) int s_=a<b?1:-1,b_=b;for(int i=a;i!=b_;i+=s_)
#define SQZ(i, j, a, b) for (int i=int(a),j=int(b)-1;i<j;++i,--j)
#define SQZ_1(i, j, a, b) for (int i=int(a),j=int(b);i<=j;++i,--j)
#define REP_2(i, j, n, m) REP(i, n) REP(j, m)
#define REP_2_1(i, j, n, m) REP_1(i, n) REP_1(j, m)
#define ALL(A) A.begin(), A.end()
#define LLA(A) A.rbegin(), A.rend()
#define CPY(A, B) memcpy(A, B, sizeof(A))
#define INS(A, P, B) A.insert(A.begin() + P, B)
#define ERS(A, P) A.erase(A.begin() + P)
#define BSC(A, X) find(ALL(A), X) // != A.end()
#define CTN(T, x) (T.find(x) != T.end())
#define SZ(A) int(A.size())
#define PB push_back
#define MP(A, B) make_pair(A, B)
#define Rush int T____; RD(T____); DO(T____)
#define Display(A, n, m) { \
    REP(i, n){         \
 REP(j, m) cout << A[i][j] << " ";\
        cout << endl;                \
    }                        \
}
#define Display_1(A, n, m) {                \
    REP_1(i, n){        \
 REP_1(j, m) cout << A[i][j] << " ";\
        cout << endl;                \
    }                        \
}
#pragma comment(linker, "/STACK:36777216")
#define fi first
#define se second
typedef long long LL;
typedef double DB;
typedef unsigned UINT;
typedef unsigned long long ULL;
typedef vector<int> VI;
typedef vector<char> VC;
typedef vector<string> VS;
typedef vector<LL> VL;
typedef vector<DB> VD;
typedef set<int> SI;
typedef set<string> SS;
typedef set<LL> SL;
typedef set<DB> SD;
typedef map<int, int> MII;
typedef map<string, int> MSI;
typedef map<LL, int> MLI;
typedef map<DB, int> MDI;
typedef map<int, bool> MIB;
typedef map<string, bool> MSB;
typedef map<LL, bool> MLB;
typedef map<DB, bool> MDB;
typedef pair<int, int> PII;
typedef pair<int, bool> PIB;
typedef vector<PII> VII;
typedef vector<VI> VVI;
typedef vector<VII> VVII;
typedef set<PII> SII;
typedef map<PII, int> MPIII;
typedef map<PII, bool> MPIIB;
template<class T> inline int RD(T &);
template<class T> inline void OT(const T &);
inline LL RD(){LL x; return RD(x);}
inline char RC(char &c){scanf(" %c", &c); return c;}
inline char RC(){char c; return RC(c);}
inline DB RF(DB &x){scanf("%lf", &x); return x;}
inline DB RF(){DB x; return RF(x);}
inline char* RS(char *s){scanf("%s", s); return s;}
template<class T0, class T1> inline void RD(T0 &x0, T1 &x1){RD(x0), RD(x1);}
template<class T0, class T1, class T2> inline void RD(T0 &x0, T1 &x1, T2 &x2){RD(x0), RD(x1), RD(x2);}
template<class T0, class T1, class T2, class T3> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3){RD(x0), RD(x1), RD(x2), RD(x3);}
template<class T0, class T1, class T2, class T3, class T4> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5), RD(x6);}
template<class T0, class T1> inline void OT(T0 &x0, T1 &x1){OT(x0), OT(x1);}
template<class T0, class T1, class T2> inline void OT(T0 &x0, T1 &x1, T2 &x2){OT(x0), OT(x1), OT(x2);}
template<class T0, class T1, class T2, class T3> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3){OT(x0), OT(x1), OT(x2), OT(x3);}
template<class T0, class T1, class T2, class T3, class T4> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5), OT(x6);}
inline void RC(char &a, char &b){RC(a), RC(b);}
inline void RC(char &a, char &b, char &c){RC(a), RC(b), RC(c);}
inline void RF(DB &a, DB &b){RF(a), RF(b);}
inline void RF(DB &a, DB &b, DB &c){RF(a), RF(b), RF(c);}
inline void RS(char *s1, char *s2){RS(s1), RS(s2);}
inline void RS(char *s1, char *s2, char *s3){RS(s1), RS(s2), RS(s3);}
template<class T> inline void RST(T &A){memset(A, 0, sizeof(A));}
template<class T0, class T1> inline void RST(T0 &A0, T1 &A1){RST(A0), RST(A1);}
template<class T0, class T1, class T2> inline void RST(T0 &A0, T1 &A1, T2 &A2){RST(A0), RST(A1), RST(A2);}
template<class T0, class T1, class T2, class T3> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3){RST(A0), RST(A1), RST(A2), RST(A3);}
template<class T0, class T1, class T2, class T3, class T4> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5), RST(A6);}
template<class T> inline void FLC(T &A, int x){memset(A, x, sizeof(A));}
template<class T0, class T1> inline void FLC(T0 &A0, T1 &A1, int x){FLC(A0, x), FLC(A1, x);}
template<class T0, class T1, class T2> inline void FLC(T0 &A0, T1 &A1, T2 &A2){FLC(A0), FLC(A1), FLC(A2);}
template<class T0, class T1, class T2, class T3> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3){FLC(A0), FLC(A1), FLC(A2), FLC(A3);}
template<class T0, class T1, class T2, class T3, class T4> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4), FLC(A5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4), FLC(A5), FLC(A6);}
template<class T> inline void CLR(priority_queue<T, vector<T>, less<T> > &Q){while (!Q.empty()) Q.pop();}
template<class T> inline void CLR(priority_queue<T, vector<T>, greater<T> > &Q){while (!Q.empty()) Q.pop();}
template<class T> inline void CLR(T &A){A.clear();}
template<class T0, class T1> inline void CLR(T0 &A0, T1 &A1){CLR(A0), CLR(A1);}
template<class T0, class T1, class T2> inline void CLR(T0 &A0, T1 &A1, T2 &A2){CLR(A0), CLR(A1), CLR(A2);}
template<class T0, class T1, class T2, class T3> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3){CLR(A0), CLR(A1), CLR(A2), CLR(A3);}
template<class T0, class T1, class T2, class T3, class T4> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5), CLR(A6);}
template<class T> inline void CLR(T &A, int n){REP(i, n) CLR(A[i]);}
template<class T> inline void SRT(T &A){sort(ALL(A));}
template<class T, class C> inline void SRT(T &A, C B){sort(ALL(A), B);}
const int MOD = 1000000007;
const int INF = 0x3f3f3f3f;
const DB EPS = 1e-6;
const DB OO = 1e15;
const DB PI = acos(-1.0); //M_PI;
template<class T> inline void checkMin(T &a,const T b){if (b<a) a=b;}
template<class T> inline void checkMax(T &a,const T b){if (b>a) a=b;}
template <class T, class C> inline void checkMin(T& a, const T b, C c){if (c(b,a)) a = b;}
template <class T, class C> inline void checkMax(T& a, const T b, C c){if (c(a,b)) a = b;}
template<class T> inline T min(T a, T b, T c){return min(min(a, b), c);}
template<class T> inline T max(T a, T b, T c){return max(max(a, b), c);}
template<class T> inline T min(T a, T b, T c, T d){return min(min(a, b), min(c, d));}
template<class T> inline T max(T a, T b, T c, T d){return max(max(a, b), max(c, d));}
template<class T> inline T sqr(T a){return a*a;}
template<class T> inline T cub(T a){return a*a*a;}
inline int Ceil(int x, int y){return (x - 1) / y + 1;}
inline bool _1(int x, int i){return x&1<<i;}
inline bool _1(LL x, int i){return x&1LL<<i;}
inline LL _1(int i){return 1LL<<i;}
inline LL _U(int i){return _1(i) - 1;};
template<class T> inline T low_bit(T x) {return x & -x;}
template<class T> inline T high_bit(T x) {T p = low_bit(x);while (p != x) x -= p, p = low_bit(x);return p;}
template<class T> inline T cover_bit(T x){T p = 1; while (p < x) p <<= 1;return p;}
inline int count_bits(int x){
 x = (x & 0x55555555) + ((x & 0xaaaaaaaa) >> 1);
 x = (x & 0x33333333) + ((x & 0xcccccccc) >> 2);
 x = (x & 0x0f0f0f0f) + ((x & 0xf0f0f0f0) >> 4);
 x = (x & 0x00ff00ff) + ((x & 0xff00ff00) >> 8);
 x = (x & 0x0000ffff) + ((x & 0xffff0000) >> 16);
 return x;
}
inline int count_bits(LL x){
 x = (x & 0x5555555555555555LL) + ((x & 0xaaaaaaaaaaaaaaaaLL) >> 1);
 x = (x & 0x3333333333333333LL) + ((x & 0xccccccccccccccccLL) >> 2);
 x = (x & 0x0f0f0f0f0f0f0f0fLL) + ((x & 0xf0f0f0f0f0f0f0f0LL) >> 4);
 x = (x & 0x00ff00ff00ff00ffLL) + ((x & 0xff00ff00ff00ff00LL) >> 8);
 x = (x & 0x0000ffff0000ffffLL) + ((x & 0xffff0000ffff0000LL) >> 16);
 x = (x & 0x00000000ffffffffLL) + ((x & 0xffffffff00000000LL) >> 32);
 return x;
}
inline int reverse_bits(int x){
 x = ((x >> 1) & 0x55555555) | ((x << 1) & 0xaaaaaaaa);
 x = ((x >> 2) & 0x33333333) | ((x << 2) & 0xcccccccc);
 x = ((x >> 4) & 0x0f0f0f0f) | ((x << 4) & 0xf0f0f0f0);
 x = ((x >> 8) & 0x00ff00ff) | ((x << 8) & 0xff00ff00);
 x = ((x >>16) & 0x0000ffff) | ((x <<16) & 0xffff0000);
 return x;
}
inline LL reverse_bits(LL x){
 x = ((x >> 1) & 0x5555555555555555LL) | ((x << 1) & 0xaaaaaaaaaaaaaaaaLL);
 x = ((x >> 2) & 0x3333333333333333LL) | ((x << 2) & 0xccccccccccccccccLL);
 x = ((x >> 4) & 0x0f0f0f0f0f0f0f0fLL) | ((x << 4) & 0xf0f0f0f0f0f0f0f0LL);
 x = ((x >> 8) & 0x00ff00ff00ff00ffLL) | ((x << 8) & 0xff00ff00ff00ff00LL);
 x = ((x >>16) & 0x0000ffff0000ffffLL) | ((x <<16) & 0xffff0000ffff0000LL);
 x = ((x >>32) & 0x00000000ffffffffLL) | ((x <<32) & 0xffffffff00000000LL);
 return x;
}
inline void INC(int &a, int b){a += b; if (a >= MOD) a -= MOD;}
inline int sum(int a, int b){a += b; if (a >= MOD) a -= MOD; return a;}
inline void DEC(int &a, int b){a -= b; if (a < 0) a += MOD;}
inline int dff(int a, int b){a -= b; if (a < 0) a += MOD; return a;}
inline void MUL(int &a, int b){a = (LL)a * b % MOD;}
inline int pdt(int a, int b){return (LL)a * b % MOD;}
inline int sum(int a, int b, int c){return sum(sum(a, b), c);}
inline int sum(int a, int b, int c, int d){return sum(sum(a, b), sum(c, d));}
inline int pdt(int a, int b, int c){return pdt(pdt(a, b), c);}
inline int pdt(int a, int b, int c, int d){return pdt(pdt(pdt(a, b), c), d);}
inline int pow(int a, LL b){
 int c(1); while (b){
 if (b&1) MUL(c, a);
 MUL(a, a), b >>= 1;
 }
 return c;
}
template<class T> inline T pow(T a, LL b){
 T c(1); while (b){
 if (b&1) c *= a;
 a *= a, b >>= 1;
 }
 return c;
}
inline int _I(int b){
 int a = MOD, x1 = 0, x2 = 1, q;
 while (true){
 q = a / b, a %= b;
 if (!a) return (x2 + MOD) % MOD;
 DEC(x1, pdt(q, x2));
 q = b / a, b %= a;
 if (!b) return (x1 + MOD) % MOD;
 DEC(x2, pdt(q, x1));
 }
}
inline void DIA(int &a, int b){MUL(a, _I(b));}
inline int qtt(int a, int b){return pdt(a, _I(b));}
inline int phi(int n){
 int res = n; for (int i=2;sqr(i)<=n;++i) if (!(n%i)){
 DEC(res, qtt(res, i));
 do{n /= i;} while(!(n%i));
 }
 if (n != 1)
 DEC(res, qtt(res, n));
 return res;
}
struct Po; struct Line; struct Seg;
inline int sgn(DB x){return x < -EPS ? -1 : x > EPS;}
inline int sgn(DB x, DB y){return sgn(x - y);}
struct Po{
 DB x, y;
 Po(DB _x = 0, DB _y = 0):x(_x), y(_y){}
 friend istream& operator >>(istream& in, Po &p){return in >> p.x >> p.y;}
 friend ostream& operator <<(ostream& out, Po p){return out << "(" << p.x << ", " << p.y << ")";}
 friend bool operator ==(Po, Po);
 friend bool operator !=(Po, Po);
 friend Po operator +(Po, Po);
 friend Po operator -(Po, Po);
 friend Po operator *(Po, DB);
 friend Po operator /(Po, DB);
 bool operator < (const Po &rhs) const{return sgn(x, rhs.x) < 0 || sgn(x, rhs.x) == 0 && sgn(y, rhs.y) < 0;}
 Po operator-() const{return Po(-x, -y);}
 Po& operator +=(Po rhs){x += rhs.x, y += rhs.y; return *this;}
 Po& operator -=(Po rhs){x -= rhs.x, y -= rhs.y; return *this;}
 Po& operator *=(DB k){x *= k, y *= k; return *this;}
 Po& operator /=(DB k){x /= k, y /= k; return *this;}
 DB length_sqr(){return sqr(x) + sqr(y);}
 DB length(){return sqrt(length_sqr());}
 DB atan(){
 return atan2(y, x);
 }
 void input(){
 scanf("%lf %lf", &x, &y);
 }
};
bool operator ==(Po a, Po b){return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0;}
bool operator !=(Po a, Po b){return sgn(a.x - b.x) != 0 || sgn(a.y - b.y) != 0;}
Po operator +(Po a, Po b){return Po(a.x + b.x, a.y + b.y);}
Po operator -(Po a, Po b){return Po(a.x - b.x, a.y - b.y);}
Po operator *(Po a, DB k){return Po(a.x * k, a.y * k);}
Po operator *(DB k, Po a){return a * k;}
Po operator /(Po a, DB k){return Po(a.x / k, a.y / k);}
struct Line{
 Po a, b;
 Line(Po _a = Po(), Po _b = Po()):a(_a), b(_b){}
 Line(DB x0, DB y0, DB x1, DB y1):a(Po(x0, y0)), b(Po(x1, y1)){}
 Line(Seg);
 friend ostream& operator <<(ostream& out, Line p){return out << p.a << "-" << p.b;}
};
struct Seg{
 Po a, b;
 Seg(Po _a = Po(), Po _b = Po()):a(_a), b(_b){}
 Seg(DB x0, DB y0, DB x1, DB y1):a(Po(x0, y0)), b(Po(x1, y1)){}
 Seg(Line l);
 friend ostream& operator <<(ostream& out, Seg p){return out << p.a << "-" << p.b;}
 DB length(){return (b - a).length();}
};
Line::Line(Seg l):a(l.a), b(l.b){}
Seg::Seg(Line l):a(l.a), b(l.b){}
#define innerProduct dot
#define scalarProduct dot
#define dotProduct dot
#define outerProduct det
#define crossProduct det
inline DB dot(DB x1, DB y1, DB x2, DB y2){return x1 * x2 + y1 * y2;}
inline DB dot(Po a, Po b){return dot(a.x, a.y, b.x, b.y);}
inline DB dot(Po p0, Po p1, Po p2){return dot(p1 - p0, p2 - p0);}
inline DB dot(Line l1, Line l2){return dot(l1.b - l1.a, l2.b - l2.a);}
inline DB det(DB x1, DB y1, DB x2, DB y2){return x1 * y2 - x2 * y1;}
inline DB det(Po a, Po b){return det(a.x, a.y, b.x, b.y);}
inline DB det(Po p0, Po p1, Po p2){return det(p1 - p0, p2 - p0);}
inline DB det(Line l1, Line l2){return det(l1.b - l1.a, l2.b - l2.a);}
template<class T1, class T2> inline DB dist(T1 x, T2 y){return sqrt(dist_sqr(x, y));}
inline DB dist_sqr(Po a, Po b){return sqr(a.x - b.x) + sqr(a.y - b.y);}
inline DB dist_sqr(Po p, Line l){Po v0 = l.b - l.a, v1 = p - l.a; return sqr(fabs(det(v0, v1))) / v0.length_sqr();}
inline DB dist_sqr(Po p, Seg l){
 Po v0 = l.b - l.a, v1 = p - l.a, v2 = p - l.b;
 if (sgn(dot(v0, v1)) * sgn(dot(v0, v2)) <= 0) return dist_sqr(p, Line(l));
 else return min(v1.length_sqr(), v2.length_sqr());
}
inline DB dist_sqr(Line l, Po p){return dist_sqr(p, l);}
inline DB dist_sqr(Seg l, Po p){return dist_sqr(p, l);}
inline DB dist_sqr(Line l1, Line l2){
 if (sgn(det(l1, l2)) != 0) return 0;
 return dist_sqr(l1.a, l2);
}
inline DB dist_sqr(Line l1, Seg l2){
 Po v0 = l1.b - l1.a, v1 = l2.a - l1.a, v2 = l2.b - l1.a; DB c1 = det(v0, v1), c2 = det(v0, v2);
 return sgn(c1) != sgn(c2) ? 0 : sqr(min(fabs(c1), fabs(c2))) / v0.length_sqr();
}
bool isIntersect(Seg l1, Seg l2){
 return
 min(l1.a.x, l1.b.x) <= max(l2.a.x, l2.b.x) &&
 min(l2.a.x, l2.b.x) <= max(l1.a.x, l1.b.x) &&
 min(l1.a.y, l1.b.y) <= max(l2.a.y, l2.b.y) &&
 min(l2.a.y, l2.b.y) <= max(l1.a.y, l1.b.y) &&
 sgn( det(l1.a, l2.a, l2.b) ) * sgn( det(l1.b, l2.a, l2.b) ) <= 0 &&
 sgn( det(l2.a, l1.a, l1.b) ) * sgn( det(l2.b, l1.a, l1.b) ) <= 0;
}
inline DB dist_sqr(Seg l1, Seg l2){
 if (isIntersect(l1, l2)) return 0;
 else return min(dist_sqr(l1.a, l2), dist_sqr(l1.b, l2), dist_sqr(l2.a, l1), dist_sqr(l2.b, l1));
}
inline bool isOnExtremePoint(const Po &p, const Seg &l){
 return p == l.a || p == l.b;
}
inline bool isOnseg(const Po &p, const Seg &l){
 return sgn(det(p, l.a, l.b)) == 0 &&
 sgn(l.a.x, p.x) * sgn(l.b.x, p.x) <= 0 && sgn(l.a.y, p.y) * sgn(l.b.y, p.y) <= 0;
}
inline Po intersect(const Line &l1, const Line &l2){
 return l1.a + (l1.b - l1.a) * (det(l2.a, l1.a, l2.b) / det(l2, l1));
}
inline Po intersect(const Po & p, const Line &l){
 return intersect(Line(p, p + Po(l.a.y - l.b.y, l.b.x - l.a.x)), l);
}
inline Po rotate(Po p, DB alpha, Po o = Po()){
 p.x -= o.x, p.y -= o .y;
 return Po(p.x * cos(alpha) - p.y * sin(alpha), p.y * cos(alpha) + p.x * sin(alpha)) + o;
}
inline int rand32(){return (bool(rand() & 1) << 30) | (rand() << 15) + rand();}
inline int random32(int l, int r){return rand32() % (r - l + 1) + l;}
inline int random(int l, int r){return rand() % (r - l + 1) + l;}
int dice(){return rand() % 6;}
bool coin(){return rand() % 2;}
template<class T> inline int RD(T &x){
 char c; for (c = getchar(); c < '0'; c = getchar()); x = c - '0'; for (c = getchar(); c >= '0'; c = getchar()) x = x * 10 + c - '0';
 return x;
}
int ____Case; template<class T> inline void OT(const T &x){
 printf("%d\n", x);
}
const int N = 51;
map<int, VII> R;
int end_time;
struct Elevator{
 VI req[N]; int cnt[N];
 int pos, t; bool idle, dir, anchor, open;
 void init(int _pos){RST(cnt); CLR(req, N); pos = _pos, idle = true, t = -1;}
 inline bool c0(){
 FOR(i, pos + 1, N) if (cnt[i]) return true;
 return false;
 }
 inline bool c1(){
 DWN(i, pos, 0) if (cnt[i]) return true;
 return false;
 }
 inline bool __c(const VI& req){
 return !req.empty();
 }
 inline bool _c(const VI& req){
 if (!__c(req)) return false;
 REP(i, SZ(req)) if ((req[i] > pos) ^ dir) return true;
 return false;
 }
 inline bool c2(){
 FOR(i, pos + 1, N) if (__c(req[i])) return true;
 return false;
 }
 inline bool c3(){
 DWN(i, pos, 0) if (__c(req[i])) return true;
 return false;
 }
 inline void pc(){
 ++t; REP(i, SZ(R[t])) req[R[t][i].fi].PB(R[t][i].se);
 }
 inline void pt(){
 printf("%02d:%02d ", t / 60, t % 60);
 }
 void op(){
 pt(); puts("The elevator door is opening."); open = true;
 pc();
 }
 void cz(){
 pt(); puts("The elevator door is closing."); open = false;
 pc();
 }
 void mv(){
 if (anchor){
 pt(); if (!dir) printf("The elevator starts to move up from floor %d.\n", pos);
 else printf("The elevator starts to move down from floor %d.\n", pos);
 anchor = false;
 }
 if (dir) --pos; else ++pos;
 pc();
 }
 inline void rt(bool _dir){
 if (idle) idle = false, anchor = true;
 dir = _dir, mv();
 }
 void stop(){
 pt(); printf("The elevator stops at floor %d.\n", pos);
 anchor = true;
 }
 void enter(){
 if (!open) op(); if (idle == true) idle = false, dir = false, dir = !_c(req[pos]), anchor = true; pt(); int t = 0;
 ECH(it, req[pos]) if ((*it > pos) ^ dir) ++cnt[*it], req[pos].erase(it--), ++t;
 printf("%d people enter the elevator.\n", t); pc();
 }
 void leave(){
 if (!open) op();
 pt(); printf("%d people leave the elevator.\n", cnt[pos]);
 cnt[pos] = 0; pc();
 if (!dir && !c0() && !c2()) dir = true;
 else if (dir && !c1() && !c3()) dir = false;
 }
 void run(){
 pc(); while (true){
 if (idle){
 if (__c(req[pos])) enter();
 else if (c2()) rt(0);
 else if (c3()) rt(1);
 else if (t <= end_time) pc();
 else break;
 }else if (anchor){
 if (cnt[pos]) leave();
 else if (_c(req[pos])) enter();
 else if (open) cz();
 else{
 if (dir){
 if (c1() || c3()) rt(1);
 else if (c0() || c2()) rt(0);
 }
 else {
 if (c0() || c2()) rt(0);
 else if (c1() || c3()) rt(1);
 }
 if (anchor) idle = true;
 }
 }else{
 if (cnt[pos] || _c(req[pos])) stop();
 else {
 if (dir && !c1() && !c3()) stop(), idle = true;
 else if (!dir && !c0() && !c2()) stop(), idle = true;
 else mv();
 }
 }
 }
 puts("");
 }
} E;
int main(){
 Rush{
 printf("Case %d:\n", ++____Case);
 E.init(RD()); CLR(R); end_time = 0; Rush{
 int time, from, to; RD(time, from, to);
 R[time].PB(MP(from, to));
 checkMax(end_time, time);
 }
 E.run();
 }
}