// #pragma GCC target ("avx")
// #pragma GCC optimize("Ofast")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")

#ifdef MARC_LOCAL
#define _GLIBCXX_DEBUG
#endif

#ifdef ATCODER
#include <atcoder/all>
using namespace atcoder;
#endif

#include <bits/stdc++.h>
using namespace std;

//#define int long long
using ll = long long; using vll = vector<ll>; using vvll = vector<vll>; using vvvll = vector<vvll>; using vvvvll = vector<vvvll>;
using vi = vector<int>; using vvi = vector<vi>; using vvvi = vector<vvi>; using vvvvi = vector<vvvi>;
using ld = long double; using vld = vector<ld>; using vvld = vector<vld>; using vd = vector<double>;
using vc = vector<char>; using vvc = vector<vc>; using vs = vector<string>;
using vb = vector<bool>; using vvb = vector<vb>; using vvvb = vector<vvb>;
using pii = pair<int, int>; using pcc = pair<char, char>; using pll = pair<ll, ll>; using pli = pair<ll, int>; using pdd = pair<double, double>; using pldld = pair<ld,ld>;
using vpii = vector<pii>; using vvpii = vector<vpii>; using vpll = vector<pll>; using vvpll = vector<vpll>; using vpldld = vector<pldld>;
using ui = unsigned int; using ull = unsigned long long;
using i128 = __int128; using f128 = __float128;
template<class T> using pqg = priority_queue<T, vector<T>, greater<T>>;
#define vec(type, name, ...) vector<type> name(__VA_ARGS__)
// #define vv(type, name, h, ...) vector<vector<type> > name(h, vector<type>(__VA_ARGS__))
// #define vvv(type, name, h, w, ...) vector<vector<vector<type> > > name(h, vector<vector<type> >(w, vector<type>(__VA_ARGS__)))
// #define vvvv(type, name, a, b, c, ...) vector<vector<vector<vector<type> > > > name(a, vector<vector<vector<type> > >(b, vector<vector<type> >(c, vector<type>(__VA_ARGS__))))

#define overload4(a,b,c,d,name,...) name
#define rep1(n) for(ll i = 0; i < (ll)n; i++)
#define rep2(i,n) for(ll i = 0; i < (ll)n; i++)
#define rep3(i,a,b) for (ll i = a; i < (ll)b; i++)
#define rep4(i,a,b,c) for (ll i = a; i < (ll)b; i += (ll)c)
#define rep(...) overload4(__VA_ARGS__,rep4,rep3,rep2,rep1)(__VA_ARGS__)
#define repback1(n) for (ll i = (ll)n-1; i >= 0; i--)
#define repback2(i,n) for (ll i = (ll)n-1; i >= 0; i--)
#define repback3(i,a,b) for (ll i = (ll)b-1; i >= (ll)a; i--)
#define repback4(i,a,b,c) for (ll i = (ll)b-1; i >= (ll)a; i -= (ll)c)
#define repback(...) overload4(__VA_ARGS__,repback4,repback3,repback2,repback1)(__VA_ARGS__)
#define all(x) (x).begin(), (x).end()
#define include(y, x, H, W) (0 <= (y) && (y) < (H) && 0 <= (x) && (x) < (W))
#define inrange(x, down, up) ((down) <= (x) && (x) <= (up))
#define square(x) (x) * (x)
#define pb push_back
#define eb emplace_back
#define fir first
#define sec second
#define equals(a,b) (fabs((a) - (b)) < EPS)
#define TIMER_START TIME_START = clock()
#ifdef MARC_LOCAL
// https://trap.jp/post/1224/
#define debug1(x) cout << "debug: " << (#x) << ": " << (x) << endl
#define debug2(x, y) cout << "debug: " << (#x) << ": " << (x) << ", " << (#y) << ": " << (y) << endl
#define debug3(x, y, z) cout << "debug: " << (#x) << ": " << (x) << ", " << (#y) << ": " << (y) << ", " << (#z) << ": " << (z) << endl
#define debug4(x, y, z, w) cout << "debug: " << (#x) << ": " << (x) << ", " << (#y) << ": " << (y) << ", " << (#z) << ": " << (z) << ", " << (#w) << ": " << (w) << endl
#define debug5(x, y, z, w, v) cout << "debug: " << (#x) << ": " << (x) << ", " << (#y) << ": " << (y) << ", " << (#z) << ": " << (z) << ", " << (#w) << ": " << (w) << ", " << (#v) << ": " << (v) << endl
#define debug6(x, y, z, w, v, u) cout << "debug: " << (#x) << ": " << (x) << ", " << (#y) << ": " << (y) << ", " << (#z) << ": " << (z) << ", " << (#w) << ": " << (w) << ", " << (#v) << ": " << (v) << ", " << (#u) << ": " << (u) << endl
#define overload6(a, b, c, d, e, f, g,...) g
#define debug(...) overload6(__VA_ARGS__, debug6, debug5, debug4, debug3, debug2, debug1)(__VA_ARGS__)
#define debuga cerr << "a" << endl
#define debugnl cout << endl
#define Case(i) cout << "Case #" << (i) << ": "
#define TIMECHECK cerr << 1000.0 * static_cast<double>(clock() - TIME_START) / CLOCKS_PER_SEC << "ms" << endl
#define LOCAL 1
#else
#define debug1(x) void(0)
#define debug2(x, y) void(0)
#define debug3(x, y, z) void(0)
#define debug4(x, y, z, w) void(0)
#define debug5(x, y, z, w, v) void(0)
#define debug6(x, y, z, w, v, u) void(0)
#define debug(...) void(0)
#define debuga void(0)
#define debugnl void(0)
#define Case(i) void(0)
#define TIMECHECK void(0)
#define LOCAL 0
#endif

//mt19937_64 rng(0);
mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count());
clock_t TIME_START;
const long double pi = 3.1415926535897932;
const long long INFL = 1000000000000000000ll;
const long long INFLMAX = numeric_limits< long long >::max(); // 9223372036854775807
const int INF = 1000000000;
const int INFMAX = numeric_limits< int >::max(); // 2147483647
const long double INFD = numeric_limits<ld>::infinity();
const long double EPS = 1e-9;
const int mod1 = 1000000007;
const int mod2 = 998244353;
const vi dx1 = {1,0,-1,0};
const vi dy1 = {0,1,0,-1};
const vi dx2 = {0, 1, 1, 1, 0, -1, -1, -1, 0};
const vi dy2 = {1, 1, 0, -1, -1, -1, 0, 1, 1};
constexpr char nl = '\n';
constexpr char bl = ' ';

std::ostream &operator<<(std::ostream &dest, __int128_t value) { std::ostream::sentry s(dest); if (s) {__uint128_t tmp = value < 0 ? -value : value; char buffer[128]; char *d = std::end(buffer); do {--d;*d = "0123456789"[tmp % 10];tmp /= 10;} while (tmp != 0);if (value < 0) {--d;*d = '-';}int len = std::end(buffer) - d;if (dest.rdbuf()->sputn(d, len) != len) {dest.setstate(std::ios_base::badbit);}}return dest; }
__int128 parse(string &s) { __int128 ret = 0; for (int i = 0; i < (int)s.length(); i++) if ('0' <= s[i] && s[i] <= '9') ret = 10 * ret + s[i] - '0'; return ret; } // for __int128 (https://kenkoooo.hatenablog.com/entry/2016/11/30/163533)
template<typename T> ostream& operator << (ostream& os, vector<T>& vec) { os << "["; for (int i = 0; i<(int)vec.size(); i++) { os << vec[i] << (i + 1 == (int)vec.size() ? "" : ", "); } os << "]"; return os; } /// vector 出力
template<typename T, typename U> ostream& operator << (ostream& os, pair<T, U>& pair_var) { os << "(" << pair_var.first << ", " << pair_var.second << ")"; return os; } // pair 出力
template<typename T, typename U> ostream& operator << (ostream& os, map<T, U>& map_var) { os << "{"; for (auto itr = map_var.begin(); itr != map_var.end(); itr++) { os << "(" << itr->first << ", " << itr->second << ")"; itr++; if(itr != map_var.end()) os << ", "; itr--; } os << "}"; return os; } // map出力
template<typename T> ostream& operator << (ostream& os, set<T>& set_var) { os << "{"; for (auto itr = set_var.begin(); itr != set_var.end(); itr++) { os << *itr; ++itr; if(itr != set_var.end()) os << ", "; itr--; } os << "}"; return os; } /// set 出力

int popcnt(unsigned long long a){ return __builtin_popcountll(a); } // ll は 64bit対応！
int MSB1(unsigned long long x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // MSB(1), 0-indexed ( (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) )
int LSB1(unsigned long long x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } // LSB(1), 0-indexed ( (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) )
long long pow2(int n) { return 1LL << n; }
long long maskbit(int n) { return (1LL << n) - 1; }
bool bit_is1(long long x, int i) { return ((x>>i) & 1); }
string charrep(int n, char c) { return std::string(n, c); }
template<class T>void UNIQUE(T& A) {sort(all(A)); A.erase(unique(all(A)), A.end());}
template<class T>bool chmax(T& a, const T& b) { if (a < b) { a = b; return 1; } return 0; }
template<class T>bool chmin(T& a, const T& b) { if (b < a) { a = b; return 1; } return 0; }
template<class T, class U> bool chmin(T& a, const U& b){ return chmin(a, (T)b); }
template<class T, class U> bool chmax(T& a, const U& b){ return chmax(a, (T)b); }
void YESNO(bool b) {if(b){cout<<"YES"<<'\n';} else{cout<<"NO"<<'\n';}} void YES() {YESNO(true);} void NO() {YESNO(false);}
void yesno(bool b) {if(b){cout<<"yes"<<'\n';} else{cout<<"no"<<'\n';}} void yes() {yesno(true);} void no() {yesno(false);}

void POSIMPOS(bool b) {if(b){cout<<"POSSIBLE"<<'\n';} else{cout<<"IMPOSSIBLE"<<'\n';}}
void PosImpos(bool b) {if(b){cout<<"Possible"<<'\n';} else{cout<<"Impossible"<<'\n';}}
void posimpos(bool b) {if(b){cout<<"possible"<<'\n';} else{cout<<"impossible"<<'\n';}}
void FIRSEC(bool b) {if(b){cout<<"FIRST"<<'\n';} else{cout<<"SECOND"<<'\n';}}
void firsec(bool b) {if(b){cout<<"first"<<'\n';} else{cout<<"second"<<'\n';}}
void FirSec(bool b) {if(b){cout<<"First"<<'\n';} else{cout<<"Second"<<'\n';}}
void AliBob(bool b) {if(b){cout<<"Alice"<<'\n';} else{cout<<"Bob"<<'\n';}}
void TakAok(bool b) {if(b){cout<<"Takahashi"<<'\n';} else{cout<<"Aoki"<<'\n';}}
int GetTime() {return 1000.0*static_cast<double>(clock() - TIME_START) / CLOCKS_PER_SEC;}
ll myRand(ll B) {return (unsigned long long)rng() % B;}
template<typename T> void print(const T& x, const char endch = '\n') { cout << x << endch; }

template<class T> T ceil_div(T x, T y) { assert(y); return (x > 0 ? (x + y - 1) / y : x / y); }
template<class T> T floor_div(T x, T y) { assert(y); return (x > 0 ? x / y : (x - y + 1) / y); }
template<class T> pair<T, T> divmod(T x, T y) { T q = floor_div(x, y); return {q, x - q * y}; } /// (q, r) s.t. x = q*y + r 
ll GCD(ll a, ll b) { if(a < b) swap(a, b); if(b == 0) return a; if(a%b == 0) return b; else return GCD(b, a%b); }
ll LCM(ll a, ll b) { assert(GCD(a,b) != 0); return a / GCD(a, b) * b; }
ll MOD(ll &x, const ll P) { ll ret = x%P; if(ret < 0) ret += P; return x = ret; } /// x % P を非負整数に直す
ll mpow(ll x, ll n, const ll mod) { x %= mod; ll ret = 1; while(n > 0) { if(n & 1) ret = ret * x % mod; x = x * x % mod; n >>= 1; } return ret; } /// x^n % mod を計算
ll lpow(ll x, ll n) { ll ret = 1; while(n > 0){ if(n & 1) ret = ret * x; x = x * x; n >>= 1; } return ret; } /// x^nを計算
string toBinary(ll n) { if(n == 0) return "0"; assert(n > 0); string ret; while (n != 0){ ret += ( (n & 1) == 1 ? '1' : '0' ); n >>= 1; } reverse(ret.begin(), ret.end()); return ret; } /// 10進数(long long) -> 2進数(string)への変換
ll toDecimal(string S) { ll ret = 0; for(int i = 0; i < (int)S.size(); i++){ ret *= 2LL; if(S[i] == '1') ret += 1; } return ret; } /// 2進数(string)　→　10進数(long long)への変換
int ceil_pow2(ll n) { int x = 0; while ((1ll << x) < n) x++; return x;} /// return minimum non-negative `x` s.t. `n <= 2**x`
int floor_pow2(ll n) { int x = 0; while ((1ll << (x+1)) <= n) x++; return x;} /// return maximum non-negative `x` s.t. `n >= 2**x`




#include <unistd.h>

// yosupo-san arigatou
namespace yosupo_IO {

template <class T>
using is_signed_int128 =
    typename std::conditional<is_same<T, __int128_t>::value
                                  || is_same<T, __int128>::value,
                              true_type, false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<is_same<T, __uint128_t>::value
                                  || is_same<T, unsigned __int128>::value,
                              true_type, false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<is_same<T, __int128_t>::value, __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral =
    typename std::conditional<std::is_integral<T>::value
                                  || is_signed_int128<T>::value
                                  || is_unsigned_int128<T>::value,
                              true_type, false_type>::type;

template <class T>
using is_signed_int =
    typename std::conditional<(is_integral<T>::value
                               && std::is_signed<T>::value)
                                  || is_signed_int128<T>::value,
                              true_type, false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value
                               && std::is_unsigned<T>::value)
                                  || is_unsigned_int128<T>::value,
                              true_type, false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value, make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value, std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

template <class T>
using is_integral_t = enable_if_t<is_integral<T>::value>;

template <class T>
using is_signed_int_t = enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = enable_if_t<is_unsigned_int<T>::value>;

namespace detail {
template <typename T, decltype(&T::is_modint) = &T::is_modint>
std::true_type check_value(int);
template <typename T>
std::false_type check_value(long);
} // namespace detail

template <typename T>
struct is_modint : decltype(detail::check_value<T>(0)) {};

template <typename T>
using is_modint_t = enable_if_t<is_modint<T>::value>;

template <class T>
using to_unsigned_t = typename to_unsigned<T>::type;

struct Scanner {
public:
  Scanner(const Scanner &) = delete;
  Scanner &operator=(const Scanner &) = delete;

  Scanner(FILE *fp) : fd(fileno(fp)) { line[0] = 127; }

  void read() {}
  template <class H, class... T>
  void read(H &h, T &... t) {
    bool f = read_single(h);
    assert(f);
    read(t...);
  }

  int read_unsafe() { return 0; }
  template <class H, class... T>
  int read_unsafe(H &h, T &... t) {
    bool f = read_single(h);
    if (!f) return 0;
    return 1 + read_unsafe(t...);
  }

  int close() { return ::close(fd); }

private:
  static constexpr int SIZE = 1 << 15;

  int fd = -1;
  array<char, SIZE + 1> line;
  int st = 0, ed = 0;
  bool eof = false;

  bool read_single(string &ref) {
    if (!skip_space()) return false;
    ref = "";
    while (true) {
      char c = top();
      if (c <= ' ') break;
      ref += c;
      st++;
    }
    return true;
  }
  bool read_single(double &ref) {
    string s;
    if (!read_single(s)) return false;
    ref = std::stod(s);
    return true;
  }

  template <class T, enable_if_t<is_same<T, char>::value> * = nullptr>
  bool read_single(T &ref) {
    if (!skip_space<50>()) return false;
    ref = top();
    st++;
    return true;
  }

  template <class T, is_signed_int_t<T> * = nullptr,
            enable_if_t<!is_same<T, char>::value> * = nullptr>
  bool read_single(T &sref) {
    using U = to_unsigned_t<T>;
    if (!skip_space<50>()) return false;
    bool neg = false;
    if (line[st] == '-') {
      neg = true;
      st++;
    }
    U ref = 0;
    do { ref = 10 * ref + (line[st++] & 0x0f); } while (line[st] >= '0');
    sref = neg ? -ref : ref;
    return true;
  }
  template <class U, is_unsigned_int_t<U> * = nullptr,
            enable_if_t<!is_same<U, char>::value> * = nullptr>
  bool read_single(U &ref) {
    if (!skip_space<50>()) return false;
    ref = 0;
    do { ref = 10 * ref + (line[st++] & 0x0f); } while (line[st] >= '0');
    return true;
  }

  bool reread() {
    if (ed - st >= 50) return true;
    if (st > SIZE / 2) {
      std::memmove(line.data(), line.data() + st, ed - st);
      ed -= st;
      st = 0;
    }
    if (eof) return false;
    auto u = ::read(fd, line.data() + ed, SIZE - ed);
    if (u == 0) {
      eof = true;
      line[ed] = '\0';
      u = 1;
    }
    ed += int(u);
    line[ed] = char(127);
    return true;
  }

  char top() {
    if (st == ed) {
      bool f = reread();
      assert(f);
    }
    return line[st];
  }

  template <int TOKEN_LEN = 0>
  bool skip_space() {
    while (true) {
      while (line[st] <= ' ') st++;
      if (ed - st > TOKEN_LEN) return true;
      if (st > ed) st = ed;
      for (auto i = st; i < ed; i++) {
        if (line[i] <= ' ') return true;
      }
      if (!reread()) return false;
    }
  }
};

struct Printer {
public:
  template <char sep = ' ', bool F = false>
  void write() {}
  template <char sep = ' ', bool F = false, class H, class... T>
  void write(const H &h, const T &... t) {
    if (F) write_single(sep);
    write_single(h);
    write<true>(t...);
  }
  template <char sep = ' ', class... T>
  void writeln(const T &... t) {
    write<sep>(t...);
    write_single('\n');
  }

  Printer(FILE *_fp) : fd(fileno(_fp)) {}
  ~Printer() { flush(); }

  int close() {
    flush();
    return ::close(fd);
  }

  void flush() {
    if (pos) {
      auto res = ::write(fd, line.data(), pos);
      assert(res != -1);
      pos = 0;
    }
  }

private:
  static array<array<char, 2>, 100> small;
  static array<unsigned long long, 20> tens;

  static constexpr size_t SIZE = 1 << 15;
  int fd;
  array<char, SIZE> line;
  size_t pos = 0;
  std::stringstream ss;

  template <class T, enable_if_t<is_same<char, T>::value> * = nullptr>
  void write_single(const T &val) {
    if (pos == SIZE) flush();
    line[pos++] = val;
  }

  template <class T, is_signed_int_t<T> * = nullptr,
            enable_if_t<!is_same<char, T>::value> * = nullptr>
  void write_single(const T &val) {
    using U = to_unsigned_t<T>;
    if (val == 0) {
      write_single('0');
      return;
    }
    if (pos > SIZE - 50) flush();
    U uval = val;
    if (val < 0) {
      write_single('-');
      uval = -uval;
    }
    write_unsigned(uval);
  }

  template <class T, is_modint_t<T> * = nullptr>
  void write_single(const T &val) {
    write_single(val.val);
  }

  static int bsr(unsigned int n) {
    return 8 * (int)sizeof(unsigned int) - 1 - __builtin_clz(n);
  }
  static int bsr(unsigned long n) {
    return 8 * (int)sizeof(unsigned long) - 1 - __builtin_clzl(n);
  }
  static int bsr(unsigned long long n) {
    return 8 * (int)sizeof(unsigned long long) - 1 - __builtin_clzll(n);
  }

  template <class U, is_unsigned_int_t<U> * = nullptr>
  void write_single(U uval) {
    if (uval == 0) {
      write_single('0');
      return;
    }
    if (pos > SIZE - 50) flush();

    write_unsigned(uval);
  }

  template <class U, is_unsigned_int_t<U> * = nullptr>
  static int calc_len(U x) {
    int i = (bsr(1ULL * x) * 3 + 3) / 10;
    if (x < tens[i])
      return i;
    else
      return i + 1;
  }

  template <class U, is_unsigned_int_t<U> * = nullptr,
            enable_if_t<2 >= sizeof(U)> * = nullptr>
  void write_unsigned(U uval) {
    size_t len = calc_len(uval);
    pos += len;

    char *ptr = line.data() + pos;
    while (uval >= 100) {
      ptr -= 2;
      memcpy(ptr, small[uval % 100].data(), 2);
      uval /= 100;
    }
    if (uval >= 10) {
      memcpy(ptr - 2, small[uval].data(), 2);
    } else {
      *(ptr - 1) = char('0' + uval);
    }
  }

  template <class U, is_unsigned_int_t<U> * = nullptr,
            enable_if_t<4 == sizeof(U)> * = nullptr>
  void write_unsigned(U uval) {
    array<char, 8> buf;
    memcpy(buf.data() + 6, small[uval % 100].data(), 2);
    memcpy(buf.data() + 4, small[uval / 100 % 100].data(), 2);
    memcpy(buf.data() + 2, small[uval / 10000 % 100].data(), 2);
    memcpy(buf.data() + 0, small[uval / 1000000 % 100].data(), 2);

    if (uval >= 100000000) {
      if (uval >= 1000000000) {
        memcpy(line.data() + pos, small[uval / 100000000 % 100].data(), 2);
        pos += 2;
      } else {
        line[pos] = char('0' + uval / 100000000);
        pos++;
      }
      memcpy(line.data() + pos, buf.data(), 8);
      pos += 8;
    } else {
      size_t len = calc_len(uval);
      memcpy(line.data() + pos, buf.data() + (8 - len), len);
      pos += len;
    }
  }

  template <class U, is_unsigned_int_t<U> * = nullptr,
            enable_if_t<8 == sizeof(U)> * = nullptr>
  void write_unsigned(U uval) {
    size_t len = calc_len(uval);
    pos += len;

    char *ptr = line.data() + pos;
    while (uval >= 100) {
      ptr -= 2;
      memcpy(ptr, small[uval % 100].data(), 2);
      uval /= 100;
    }
    if (uval >= 10) {
      memcpy(ptr - 2, small[uval].data(), 2);
    } else {
      *(ptr - 1) = char('0' + uval);
    }
  }

  template <class U, enable_if_t<is_unsigned_int128<U>::value> * = nullptr>
  void write_unsigned(U uval) {
    static array<char, 50> buf;
    size_t len = 0;
    while (uval > 0) {
      buf[len++] = char((uval % 10) + '0');
      uval /= 10;
    }
    std::reverse(buf.begin(), buf.begin() + len);
    memcpy(line.data() + pos, buf.data(), len);
    pos += len;
  }

  void write_single(const std::string &s) {
    for (char c: s) write_single(c);
  }
  void write_single(const char *s) {
    size_t len = strlen(s);
    for (size_t i = 0; i < len; i++) write_single(s[i]);
  }
  template <class T>
  void write_single(const std::vector<T> &val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write_single(' ');
      write_single(val[i]);
    }
  }
};

array<array<char, 2>, 100> Printer::small = [] {
  array<array<char, 2>, 100> table;
  for (int i = 0; i <= 99; i++) {
    table[i][1] = char('0' + (i % 10));
    table[i][0] = char('0' + (i / 10 % 10));
  }
  return table;
}();
array<ull, 20> Printer::tens = [] {
  array<ull, 20> table;
  for (int i = 0; i < 20; i++) {
    table[i] = 1;
    for (int j = 0; j < i; j++) { table[i] *= 10; }
  }
  return table;
}();

} // namespace yosupo_IO

using namespace yosupo_IO;
Scanner scanner = Scanner(stdin);
Printer printer = Printer(stdout);

// ############################
// #                          #
// #    C O D E  S T A R T    #
// #                          #
// ############################

/// 有理数型 (x/y)
// 分母分子が longlong 内に収まっても、比較の際の乗算でオーバーフローする可能性があることに注意 -> T = i128を使う
template< class T = long long>
class Fraction{
	public:
		T x,y;
		/// 約分
		void reduc(){
			int minus = 1;
			T absx = x;
            T absy = y;
            if(x < 0) minus *= -1, absx *= -1;
			if(y < 0) minus *= -1, absy *= -1;
			T g = gcd(absx, absy);
			x = minus * absx / g;
			y = absy / g;
		}
		
		Fraction(T x = 0, T y = 1): x(x), y(y) {reduc();};

        // 比較の際の乗算でオーバーフローすることがあることに注意
		bool operator<(const Fraction& right) const {return x*right.y < y*right.x;}
		bool operator<=(const Fraction& right) const {return x*right.y <= y*right.x;}
		bool operator>(const Fraction& right) const {return x*right.y > y*right.x;}
		bool operator>=(const Fraction& right) const {return x*right.y >= y*right.x;}
		bool operator==(const Fraction& right) const {return x == right.x && y == right.y;}
		Fraction operator-() const {return Fraction(-x, y);}
		Fraction& operator+=(const Fraction& v){
			x = x*v.y + y*v.x;
			y *= v.y;
			reduc();
			return *this;
		}
		Fraction operator+(const Fraction& v) const {return Fraction(*this) += v;}
		Fraction& operator-=(const Fraction& v){
			x = x*v.y - y*v.x;
			y *= v.y;
			reduc();
			return *this;
		}
		Fraction operator-(const Fraction& v) const {return Fraction(*this) -= v;}
		Fraction& operator*=(const Fraction& v){
			x *= v.x;
			y *= v.y;
			reduc();
			return *this;
		}
		Fraction operator*(const Fraction& v) const {return Fraction(*this) *= v;}
		Fraction& operator/=(const Fraction& v){
			x *= v.y;
			y *= v.x;
			reduc();
			return *this;
		}
		Fraction operator/(const Fraction& v) const {return Fraction(*this) /= v;}
		Fraction inv() const {return Fraction(y,x);}
		Fraction pow(long long t) const {
			if(t < 0) return inv().pow(-t);
			Fraction a(1, 1), d = *this;
			while(t){
				d *= d;
				if(t & 1) a *= d;
				t >>= 1;
			}
			return a;
		}

		friend ostream& operator << (ostream& os, const Fraction& v){ return os << v.x << '/' << v.y;}

    private:
        T gcd(T a, T b){
            if(a < b) swap(a, b);

            if(b == 0) return a;
            if(a%b == 0) return b;
            else return gcd(b, a%b);
        }
};

#include<boost/multiprecision/cpp_int.hpp>
using Fractionll = Fraction<boost::multiprecision::cpp_int>;
using namespace boost::multiprecision;
void YesNo(bool b) {if(b){printer.writeln("Yes");} else{printer.writeln("No");}}

// https://www.mynote-jp.com/entry/Circle-Defined-by-Three-Points
void solve() {
    int Q; scanner.read(Q);
    vi X(3);
    vi Y(3);
    rep(i,3) scanner.read(X[i], Y[i]);
    
    int a = X[0] - X[1];
    int b = Y[0] - Y[1];
    int c = X[1] - X[2];
    int d = Y[1] - Y[2];
    ll bunbo = 2 * (a * d - b * c);
    debug(bunbo);
    if(bunbo == 0) {
        vpll XY(3); rep(i,3) XY[i] = {X[i],Y[i]};
        sort(all(XY));
        ll cx = XY[1].first;
        ll cy = XY[1].second;
        ll r2 = square(X[0] - cx) + square(Y[0] - cy);

        while(Q--) {
        ll x,y; scanner.read(x, y);

        ll dist = square(x - cx) + square(y - cy);

        YesNo(dist <= r2);             
        }

        return;
    }
    ll e = square(X[0]) + square(Y[0]) - (square(X[1]) + square(Y[1]));
    ll f = square(X[1]) + square(Y[1]) - (square(X[2]) + square(Y[2]));
    
    Fractionll cx(d*e - b*f, bunbo);
    Fractionll cy(-c*e + a*f, bunbo);
    Fractionll r2 = square(Fractionll(X[0],1) - cx) + square(Fractionll(Y[0],1) - cy);
    cpp_int ir2 = r2.x / r2.y;
    debug(cx,cy,r2);

    while(Q--) {
        int x,y; scanner.read(x, y);

        Fractionll dist = square(Fractionll(x,1) - cx) + square(Fractionll(y,1) - cy);

        cpp_int id = dist.x/dist.y;
        if(id == ir2) {
            cpp_int p = dist.x * r2.y;
            cpp_int q = dist.y * r2.x;

            //YesNo(dist <= r2); 
            YesNo(p <= q);            
        }
        else YesNo(id < ir2);
    }

}



signed main() {
    cin.tie(0); ios_base::sync_with_stdio(false);
    TIMER_START;
    //cout << fixed << setprecision(15);
    

    int tt = 1;
    //cin >> tt;
    while(tt--){
        solve();
    }

    TIMECHECK;
    return 0;
}