#include "bits/stdc++.h"
#define _USE_MATH_DEFINES
#include <math.h>

using namespace std;

using LL = long long;
using VLL = std::vector<LL>;
using VVLL = std::vector<VLL>;
using VVVLL = std::vector<VVLL>;
using LD = long double;
using VLD = std::vector<LD>;
using VVLD = std::vector<VLD>;
using VVVLD = std::vector<VVLD>;
using BL = bool;
using VBL = std::vector<BL>;
using VVBL = std::vector<VBL>;
using VVVBL = std::vector<VVBL>;

template<class T> inline bool chmax(T &a, T b) { if (a < b) { a = b; return true; }return false; }
template<class T> inline bool chmin(T &a, T b) { if (a > b) { a = b; return true; }return false; }
template<class T> void bye(T a){cout << a << '\n'; exit(0);} 

//template<class T=LL> inline vector<T> cinv(LL N){ vector<T> v(N); REP(i, N)cin>>v[i]; return move(v);}
//template<class T=LL> inline vector<T> cinv(LL N){ vector<T> v(N); REP(i, N)cin>>v[i]; return move(v);}
#define REP(i,n) for(LL (i)=0;(i)<(LL)(n);(i)++)
#define REPM(i,n,m) for(LL (i)=m;(i)<(LL)(n);(i)++)
#define REPI(i,n) for(LL (i)=(n-1);(i)>=(LL)(0);(i)--)
#define ALL(v) v.begin(), v.end()
#define PRINTLN(v) {LL i=0; for(auto (a): v ){   cerr <<setw(2) <<i << " : "<<  a << endl; i++;}}
//#define PRLLSP(v) for(auto (a): v ) {cerr << a << " ";} cerr << endl


template<class T> 
void PRINTSP(vector<T> v, size_t w=3){
  for(auto (a): v ) {cerr <<setw(w)<< a << " ";} cerr << endl;
} 


struct mll{
  static LL MOD;
  LL val;
  mll(LL v = 0): val(v % MOD){ if (val < 0) val += MOD; }
  mll operator - () const { return -val; }
  mll operator + (const mll &b) const { return val + b.val; }
  mll operator - (const mll &b) const { return val - b.val; }
  mll operator * (const mll &b) const { return val * b.val; }
  mll operator / (const mll &b) const { return mll(*this) /= b; }
  mll operator + (LL b) const { return *this + mll(b); }
  mll operator - (LL b) const { return *this - mll(b); }
  mll operator * (LL b) const { return *this * mll(b); }
  friend mll operator + (LL a, const mll &b) { return b + a; }
  friend mll operator - (LL a, const mll &b) { return -b + a; }
  friend mll operator * (LL a, const mll &b) { return b * a; }
  mll &operator += (const mll &b) { val=(val+b.val)%MOD; return *this; }
  mll &operator -= (const mll &b) { val=(val+MOD-b.val)%MOD; return *this; }
  mll &operator *= (const mll &b) { val=(val*b.val)%MOD; return *this; }
  mll &operator /= (const mll &b) {
    LL c=b.val, d=MOD, u=1, v=0;
    while (d){
      LL t = c / d;
      c -= t * d; swap(c, d);
      u -= t * v; swap(u, v);
    }
    val = val * u % MOD;
    if (val < 0) val += MOD;
    return *this;
  }
  mll &operator += (LL b) { return *this += mll(b); }
  mll &operator -= (LL b) { return *this -= mll(b); }
  mll &operator *= (LL b) { return *this *= mll(b); }
  mll &operator /= (LL b) { return *this /= mll(b); }
  bool operator == (const mll &b) { return val == b.val; }
  bool operator != (const mll &b) { return val != b.val; }
  bool operator == (LL b) { return *this == mll(b); }
  bool operator != (LL b) { return *this != mll(b); }
  friend bool operator == (LL a, const mll &b) { return mll(a) == b.val; }
  friend bool operator != (LL a, const mll &b) { return mll(a) != b.val; }
  friend ostream &operator << (ostream &os, const mll &a) { return os << a.val; }
  friend istream &operator >> (istream &is, mll &a) { return is >> a.val; }
  static mll Combination(LL a, LL b){
    chmin(b, a-b);
    if (b<0) return mll(0);
    mll c = 1;
    REP(i, b) c *= a-i;
    REP(i, b) c /= i+1;
    return c;
  }
  static mll Kumiawase(LL a, LL b){
    chmin(b, a-b);
    if (b<0) return mll(0);
    return Junretu(a,b)/Kaijou(b);
  }
  static mll Kaijou(LL a){
    if (a<0)  return mll(0);
    mll c = 1;
    for(LL i=1;i<=a;i++){
      c *= i;
    }
    return c;
  }
  static mll Junretu(LL a, LL b){
    if(a<b){
      return mll(0);
    }
    mll c = 1;
    for(LL i=a;i>a-b;i--){
      c *= i;
    }

    return c;
  }
  static mll _Junretu(LL a, LL b){
    if(a<b){
      return mll(0);
    }
    return Kaijou(a)/(Kaijou(a-b));
  }
  LL get(){
    return val;
  }
};
LL mll::MOD = (LL)(1e9 + 7);// 998244353LL;//(LL)(1e9 + 7);
using vmll = std::vector<mll>;
using vvmll = std::vector<vmll>;
using vvvmll = std::vector<vvmll>;
using vvvvmll = std::vector<vvvmll>;

// a^n mod を計算する
LL modpow(LL a, LL n, LL mod) {
    long long res = 1;
    while (n > 0) {
        if (n & 1) res = res * a % mod;
        a = a * a % mod;
        n >>= 1;
    }
    return res;
}

template<typename TTT>
vector<LL> arg_sort(vector<TTT> A, bool ascend = true) {
  vector<LL> index(A.size());
  iota(index.begin(), index.end(), 0);
  if (ascend) {
    std::sort(index.begin(), index.end(), [&A](TTT i1, TTT i2) {
      return A[i1] < A[i2];
      });
  }
  else {
    std::sort(index.begin(), index.end(), [&A](TTT i1, TTT i2) {
      return A[i1] > A[i2];
      });
  }

  return index;
}

template<typename _Iterator, typename _Compare>
LL num_of_ika(_Iterator _first, _Iterator _last, _Compare key){
  return (LL)(upper_bound(_first,_last,key)-_first);
}
template<typename _Iterator, typename _Compare>
LL num_of_ookii(_Iterator _first, _Iterator _last, _Compare key){
  return (LL)(_last-upper_bound(_first,_last,key));
}
template<typename _Iterator, typename _Compare>
LL num_of_chisai(_Iterator _first, _Iterator _last, _Compare key){
  return (LL)(lower_bound(_first,_last,key) - _first);
}
template<typename _Iterator, typename _Compare>
LL num_of_ijou(_Iterator _first, _Iterator _last, _Compare key){
  return (LL)(_last-lower_bound(_first,_last,key));
}
template<typename _Iterator, typename _Compare>
LL num_of_onaji(_Iterator _first, _Iterator _last, _Compare key){
  return (LL)(upper_bound(_first,_last,key)-lower_bound(_first,_last,key));
}





#define cmax(a,b) {if(a<b){a=b;}}
#define cmin(a,b) {if(a>b){a=b;}}
#define EACH(a,A) for(auto a: A)
//#define cmax(a,b) a = (a>b ? a:b)
//#define cmin(a,b) a = (a<b ? a:b) 
//cin,cout高速化のおまじない＋桁数指定
inline VLL cinvll(LL N, LL minus = 0){
  VLL A(N);
  REP(i,N){
    cin >> A[i];
    A[i] -= minus;
  }
  return move(A);
}
inline VVLL zerosll(LL H, LL W, LL val=0){
  VVLL A(H,VLL(W,val));
  return move(A);
}
inline VVVLL zerosll3(LL H, LL W,LL C, LL val=0){
  VVVLL A(H,VVLL(W,VLL(C,val)));
  return move(A);
}


#define repi(itr, ds) for (auto itr = ds.begin(); itr != ds.end(); itr++)

// vector
template <typename T>
istream &operator>>(istream &is, vector<T> &vec) {
    for (T &x : vec) is >> x;
    return is;
}
// pair
template <typename T, typename U>
ostream &operator<<(ostream &os, pair<T, U> &pair_var) {
    os << "(" << pair_var.first << ", " << pair_var.second << ")";
    return os;
}
// vector<string>
ostream &operator<<(ostream &os, const vector<string> &vec) {
  
  LL N = to_string((LL)vec.size()).length();
  string head;
  REP(i,N+2){ head += "-";}
  REP(i,40){ head += to_string(i%10);}
  os << setw(N) << head << endl;
   
  for (LL i = 0; i < (LL)vec.size(); i++) {
    os << setw(N) << i << ": " << vec[i] << endl;
  } 
  return os;
}
// vector<LL>
ostream &operator<<(ostream &os, const vector<LL> &vec) {
  LL N = to_string((LL)vec.size()).length();
  for (LL i = 0; i < (LL)vec.size(); i++) {
    os << setw(N) << i << ": " <<vec[i] << endl;
  } 
  return os;
}
// vector<LD>
ostream &operator<<(ostream &os, const vector<LD> &vec) {
  LL N = to_string((LL)vec.size()).length();
  for (LL i = 0; i < (LL)vec.size(); i++) {
    os << setw(N) << i << ": " <<vec[i] << endl;
    //os <<vec[i] << endl;
  } 
  return os;
}
// VVLL
ostream &operator<<(ostream &os, const vector<vector<LL>> &mat) {
  LL keta = 1;
  REP(i,mat.size()){
    REP(j,mat[i].size()){
      LL x = mat[i][j];
      chmax(keta,(LL)to_string(x).length());
    }
  }
  LL N = to_string((LL)mat.size()).length();
  REP(i,mat.size()){
    os << setw(N) << i << ": " ;
    REP(j,mat[i].size()){
      LL x = mat[i][j];
      os << setw(keta+1) <<x;
    }
    os << endl;
  }  
  return os;
}

// vector
template <typename T>
ostream &operator<<(ostream &os, const vector<T> &vec) {

  for (LL i = 0; i < (LL)vec.size(); i++) {
    //os << setw(3) <<i <<" : "<< setw(5) << vec[i] << endl;
    os << setw(5) << vec[i];
  } 
  os << endl;
  return os;
}
// map
template <typename T, typename U>
ostream &operator<<(ostream &os, map<T, U> &map_var) {
    os << "{";
    repi(itr, map_var) {
        os << *itr;
        itr++;
        if (itr != map_var.end()) os << ", ";
        itr--;
    }
    os << "}";
    return os;
}
// set
template <typename T>
ostream &operator<<(ostream &os, set<T> &set_var) {
    os << "{";
    repi(itr, set_var) {
        os << *itr;
        itr++;
        if (itr != set_var.end()) os << ", ";
        itr--;
    }
    os << "}";
    
  return os;
}

#define DUMPOUT cerr
#define DUMPCOLOR ("\033[36m")

void dump_func() {
    DUMPOUT << endl;
}
template <class Head, class... Tail>
void dump_func(Head &&head, Tail &&... tail) {
    DUMPOUT << DUMPCOLOR;
    DUMPOUT << head;
    if (sizeof...(Tail) > 0) {
        DUMPOUT << ", ";
    }
    dump_func(std::move(tail)...);
    DUMPOUT << "\033[m";
}
#ifdef DEBUG_
#define DEB
#define dump(...)                                                              \
    DUMPOUT << DUMPCOLOR                                                      \
            << "" << string(#__VA_ARGS__) << ": "                            \
            << "[" << to_string(__LINE__) << ":" << __FUNCTION__ << "]"        \
            << endl                                                            \
            << "\033[m"                                                        \
            << "",                                                         \
        dump_func(__VA_ARGS__)
#else
#define DEB if (false)
#define dump(...)
#endif

struct Fast{
  Fast(){
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(std::numeric_limits<double>::max_digits10);
  }
} fast;

struct UnionFindTree {
private:
  vector<LL> UFT;
public:
  UnionFindTree(LL N) {
    UFT.resize(N);
    for (LL i = 0; i < N; i++) {
      UFT[i] = -1; 
    }
  }

  //! rootのindexを返す
  LL root(LL i) { 
    if (UFT[i] < 0) {
      return i;
    }
    else {
      LL j = root(UFT[i]);
      UFT[i] = j;
      return j;
    }
  }

  //! iとjが同じ集合かどうか
  bool same(LL i, LL j) {
    return (root(i) == root(j));
  }
  //! 統合する
  bool unite(LL i, LL j)
  {
    if(same(i,j)){
      return false;
    }

    LL root_i = root(i);
    LL root_j = root(j);
    if (root_i != root_j) {
      if (size(root_i) < size(root_j)) {
        swap(root_i, root_j);
      }
      UFT[root_i] += UFT[root_j];
      UFT[root_j] = root_i;
    }
    return true;
  }

  //! 自身が属している集合のサイズを返す 
  LL size(LL i) {
    return -UFT[root(i)];
  }

  //! 集合が何個あるかを返す
  LL get_root_num() {
    set<LL> roots;
    for (LL i = 0; i < (LL)UFT.size(); ++i) {
      roots.insert(root(i));
    }
    return (LL)roots.size();
  }

  //! 各rootに属する子を返す 
  map<LL, vector<LL>> get_root_child() {
    map<LL, vector<LL>> a;
    for (LL i = 0; i < (LL)UFT.size(); ++i) {
      LL j = root(i);
      a[j].push_back(i);
    }
    return move(a);
  }
  //! 各rootに属する子を返す 
  unordered_map<LL, vector<LL>> get_root_child2() {
    unordered_map<LL, vector<LL>> a;
    for (LL i = 0; i < (LL)UFT.size(); ++i) {
      LL j = root(i);
      a[j].push_back(i);
    }
    return move(a);
  }

  //! 標準エラーに出力する
  void print() { 
    map<LL, vector<LL>> M = get_root_child();
    for(auto m :M){
      dump(m.first, m.second);
    }
    // for (LL i = 0; i < (LL)UFT.size(); ++i) {
    //   cerr << root(i) << " ";

    // }
    cerr << endl;
  }
};
#if 0 
LL GCD(LL m, LL n) {

  if (m < n) {
    swap(m, n);
  }

  while (n != 0) {
    LL n_new = m % n;
    m = n;
    n = n_new;
  }

  return m;

}

VLL B;
VVLL T;
LL cnt=0;
void dfs(LL me,LL oya){

  for(LL i=0;i<T[me].size();i++){
    LL ko = T[me][i];
    if(ko==oya){
      continue;
    }
    cnt += min(B[me],B[ko]);
    dfs(ko,me);
  }

}
#endif


bool isPrime(LL N)
{
  if (N < 2) {
    return false;
  }
  else if (N == 2) {
    return true;
  }
  else if (N % 2 == 0) {
    return false;
  }

  LD sqrtNum = sqrt(N);
  for (LL i = 3; i <= sqrtNum; i += 2)
  {
    if (N % i == 0)
    {
      return false;
    }
  }
  return true;
}


#if 1

map< LL, LL > prime_factor(LL n) {
  map< LL,LL > ret;
  for(LL i = 2; i * i <= n; i++) {
    while(n % i == 0) {
      ret[i]++;
      n /= i;
    }
  }
  if(n != 1) ret[n] = 1;
  return ret;
}

LL GCD(LL m, LL n)
{
  if(m < n) {
    swap(m, n);
  }

  while(n != 0) {
    LL n_new = m % n;
    m = n;
    n = n_new;
  }
  return m;
}


LL LCM(LL x, LL y) { return x / GCD(x, y) * y; }

#endif

struct Furui{
  LL n;
  VLL f;
  VLL prime;

  Furui(LL n=1):n(n),f(n+1){
    f[0]=-1;
    f[1]=-1;
    for( LL i=2;i<=n;i++){
      if(f[i]){
        continue;
      }
      prime.push_back(i);
      for(LL j=i;j<=n; j+=i){
        if(f[j]==0){
          f[j] = i;
        }
      }
    }
  }
  bool isPrime(LL x){
    if(x<=1){
      return false;
    }
    return (f[x]==x);
  }
  VLL factorList(LL x){
    VLL res;
    if( x>=2 and x<=n){
      while( x!=1){
        res.push_back(f[x]);
        x /=f[x];
      }
    }
    return res;
  }
  vector<pair<LL,LL>> factor(LL x){
    VLL factors = factorList(x);
    vector<pair<LL,LL>> P;
    if(factors.size()==0){
      return P;
    }
    P.emplace_back(factors[0],0);
    for(auto p : factors){
      if(P.back().first==p){
        P.back().second++;
      }else{
        P.emplace_back(p,1);
      }
    }
    return P;
  }
};

template< typename T >
struct edge {
  int src, to;
  T cost;
  edge(int to, T cost) : src(-1), to(to), cost(cost) {}
  edge(int src, int to, T cost) : src(src), to(to), cost(cost) {}
  edge &operator=(const int &x) {
    to = x;
    return *this;
  }

  operator int() const { return to; }
};
template< typename T >
using Edges = vector< edge< T > >;
template< typename T >
using WeightedGraph = vector< Edges< T > >;
using UnWeightedGraph = vector< vector< int > >;
template< typename T >
using Matrix = vector< vector< T > >;

template< typename T >
void warshall_floyd(Matrix< T > &g, T INF) {
  for(int k = 0; k < g.size(); k++) {
    for(int i = 0; i < g.size(); i++) {
      for(int j = 0; j < g.size(); j++) {
        if(g[i][k] == INF || g[k][j] == INF) continue;
        g[i][j] = min(g[i][j], g[i][k] + g[k][j]);
      }
    }
  }
}

template< typename T >
T kruskal(Edges< T > &edges, LL V) {
  sort(begin(edges), end(edges), [](const edge< T > &a, const edge< T > &b) {
    return (a.cost < b.cost);
  });
  UnionFindTree tree(V);
  T ret = 0;
  for(auto &e : edges) {
    if(tree.unite(e.src, e.to)) ret += e.cost;
  }
  return (ret);
}




bool isOK(LL mid)
{
  return true; 
}
// 汎用的な二分探索のテンプレ
LL binary_search() {
  LL ng = 0 ;
  LL ok = 100000;

  /* ok と ng のどちらが大きいかわからないことを考慮 */
  while (abs(ok - ng) > 1) {
      LL mid = (ok + ng) / 2;

      if (isOK(mid)) ok = mid;
      else ng = mid;
  }
  return ok;
}

LL powll(LL a,LL b){
  LL c = 1LL;
  REP(i,b){
    c *= a;
  }
  return c;
}

LL knapsack(vector<LL> v,vector<LL> w,LL W) {
  LL N = (LL)v.size();
  vector<LL> dp(W+1,0);
  for (LL i=0; i<N; ++i){
    for (LL j=w[i]; j<=W; ++j){
      dp[j] = max(dp[j], dp[j-w[i]] + v[i]);
    }
  }

  return dp[W];
}


#if 0 

LL H,W;
VVLL M;
VVLL V;
LL sx;
LL sy;
LL gx;
LL gy;

void MeiroDFS(LL y,LL x){

  if(V[y][x]==1){
    return;
  }
  V[y][x]=1;

  if(y>0 and M[y-1][x]==0){
    dfs(y-1,x);
  }
  if(x>0 and M[y][x-1]==0){
    dfs(y,x-1);
  }
  if(y+1<H and M[y+1][x]==0){
    dfs(y+1,x);
  }
  if(x+1<W and M[y][x+1]==0){
    dfs(y,x+1);
  }

}

LL H,W;
VVLL M;
VVLL V;
LL sx;
LL sy;
LL gx;
LL gy;
void MeiroBFS(LL y,LL x){

  queue<tuple<LL,LL,LL>> Q;
  Q.emplace(y,x,0);
  V[y][x]=0;

  LL cnt=0;
  while(not Q.empty()){
    tuple<LL,LL,LL> q = Q.front();
    Q.pop();
    y = get<0>(q);
    x = get<1>(q);
    cnt = get<2>(q);
    if(y>0 and M[y-1][x]==0 and V[y-1][x]==-1){
      V[y-1][x]=cnt+1;
      Q.emplace(y-1,x,cnt+1);
    }
    if(x>0 and M[y][x-1]==0 and V[y][x-1]==-1){
      V[y][x-1]=cnt+1;
      Q.emplace(y,x-1,cnt+1);
    }
    if(y+1<H and M[y+1][x]==0 and V[y+1][x]==-1){
      V[y+1][x]=cnt+1;
      Q.emplace(y+1,x,cnt+1);
    }
    if(x+1<W and M[y][x+1]==0 and V[y][x+1]==-1){
      V[y][x+1]=cnt+1;
      Q.emplace(y,x+1,cnt+1);
    }
  }



}
#endif

template <typename T> vector<T> dijkstra(WeightedGraph<T> &g, int s)
{
  const auto INF = numeric_limits<T>::max();
  vector<T> dist(g.size(), INF);

  using Pi = pair<T, int>;
  priority_queue<Pi, vector<Pi>, greater<Pi>> que;
  dist[s] = 0;
  que.emplace(dist[s], s);
  while(!que.empty()) {
    T cost;
    int idx;
    tie(cost, idx) = que.top();
    que.pop();
    if(dist[idx] < cost)
      continue;
    for(auto &e : g[idx]) {
      auto next_cost = cost + e.cost;
      if(dist[e.to] <= next_cost)
        continue;
      dist[e.to] = next_cost;
      que.emplace(dist[e.to], e.to);
    }
  }
  return dist;
}

template <typename T> struct BinaryIndexedTree {
  vector<T> data;

  BinaryIndexedTree(int sz = 10) { data.assign(++sz, 0); }

  T sum(int k)
  {
    T ret = 0;
    for(++k; k > 0; k -= k & -k)
      ret += data[k];
    return (ret);
  }

  void add(int k, T x)
  {
    for(++k; k < data.size(); k += k & -k)
      data[k] += x;
  }
  void resize(int sz) { data.assign(++sz, 0); }
};

LL Combination(LL a, LL b)
{
  chmin(b, a - b);
  if(b < 0)
    return 0;
  LL c = 1;
  REP(i, b) c *= a - i;
  REP(i, b) c /= i + 1;
  return c;
}

struct IntegralImage{
  VVLL I; 
  LL H;
  LL W;
  IntegralImage(VVLL M){
    I = M;
    H = M.size();
    W = M[0].size();
    // check
    bool isOK=true;
    REP(i,H){
      if((LL)I[i].size()!=W){
        isOK=false;
      }
    }
    if(not isOK){
      cerr << "shape error :";
      REP(i,H){
        cerr << I[i].size() << " ";
      }
      cerr << endl;
      throw 0;
    }

    REP(i,H){
      REP(j,W-1){
        I[i][j+1] += I[i][j];
      }
    }
    REP(j,W){
      REP(i,H-1){
        I[i+1][j] += I[i][j];
      }
    }
  }
  LL get_sum( LL y, LL h, LL x, LL w){
    if(y<0 or x<0 or h<0 or w<0 or y+h>=H or x+w>=W){
      fprintf(stderr,"H=%lld, W=%lld, y=%lld, h=%lld, x=%lld, w=%lld\n",H,W,y,h,x,w);
      throw 1;
    }
    LL a1 = I[y+h][x+w];
    LL a2 = 0;
    if(y>0){
      a2 = I[y-1][x+w];
    }            
    LL a3 = 0;
    if(x>0){
      a3 = I[y+h][x-1];
    }            
    LL a4 = 0;
    if(y>0 && x>0){
      a4 = I[y-1][x-1];
    }
    return (a1 + a4 - a2 -a3);
  }
    LL get_sum2( LL y0, LL y1, LL x0, LL x1){
      if(y0>y1){
        swap(y0,y1);
      }
      if(x0>x1){
        swap(x0,x1);
      }
    return get_sum(y0,y1-y0 ,x0,x1-x0);
  }
  void print( ){
    REP(i,(LL)I.size()){
      PRINTSP(I[i]);
    }  
  }
};

bool isKaibun(string S){
  bool ok = true;
  LL N = (LL)S.length();
  REP(i,N){
    if(S[i]!=S[N-i-1]){
      ok = false;
      break;
    }
  }
  return ok;
}

tuple<VLL,VLL,VLL> my_dycstra(VVVLL &R,LL start, LL MOD = 0)
{
  LL N = R.size();
  LL INFINF = 1e15;
  priority_queue<VLL, VVLL, greater<VLL>> Q;
  VLL Cost(N, INFINF);
  VLL Path(N, -1);
  VLL Num(N, 0);
  Cost[start] = 0;
  Num[start] = 1;
  Q.push({0,start});
  while(!Q.empty()) {
    VLL q = Q.top();
    Q.pop();
    LL cost = q[0];
    LL from = q[1];
    //dump(from,cost);
    for(auto r : R[from]) {
      LL to = r[0];
      LL cost_to = r[1];
      //dump(to,cost_to);
      
      if(chmin(Cost[to], (cost + cost_to))) {
        Q.push({cost + cost_to, to});
        Path[to] = from;
        Num[to] = Num[from];
      }
      else if(Cost[to] == (cost + cost_to)){
        Num[to] += Num[from];
        if(MOD!=0){
          Num[to] %= MOD;
        }
      }
    }
  }
  // dump(C);
  return move(make_tuple(Cost,Path,Num));
}

VLL haba_yuusen_kyori(VVLL &V, LL n ){
  LL N = V.size();
  VLL D(N,-1);
  queue<pair<LL,LL>> Q;
  Q.push({n,0});
  while(!Q.empty()){
    auto q = Q.front();
    Q.pop();
    LL no = q.first;
    LL d  = q.second;
    D[no] = d;
    for(auto v : V[no]){
      if(D[v]!=-1){
        continue;
      }
      D[v] = d+1;
      Q.push({v,d+1});
    }    
  }
  return move(D);  
}  

struct SegmentTree{
  VLL D;
  LL N;
  LL DEFAULT;
  string mode;

  void init(LL N,string mode = "min"){
    this->mode = mode;
    if(mode=="min"){
      DEFAULT = LONG_LONG_MAX;
    }
    else if(mode=="max"){
      DEFAULT = LONG_LONG_MIN;
    }
    else if(mode=="sum"){
      DEFAULT = 0LL;
    }
    else{
      this->mode="min";
      DEFAULT = LONG_LONG_MAX;
    }
    //this->N=N;
    LL K=1;
    while(K<N){
      K*=2;
    }
    this->N = K;
    this->D.resize(2*this->N,DEFAULT);
    dump(D.size());
  }
  SegmentTree(LL N,string mode = "min"){
    init(N,mode);      
  }
  SegmentTree(VLL &A,string mode = "min"){
    LL N = A.size();
    this->init(N,mode);
    dump(D.size());
    REP(i,N){
      dump(i,A[i]);
      set(i,A[i]);
    }
    dump(N);
    build();
  }
  void set(LL i, LL x){
    D[i+N] = x;
  }
  void build(){
    for(LL i=N-1;i>0;i--){
      if(mode=="min"){
        D[i] = min({D[2*i],D[2*i+1]});
      }
      else if(mode=="max"){
        D[i] = max({D[2*i],D[2*i+1]});
      }
      else if(mode=="sum"){
        D[i] = D[2*i]+D[2*i+1];
      }
    }
  }
  void update(LL i, LL x){
    D[i+N] = x;
    LL j = i+N;
    while(j>1){
      LL k = j/2;
      if(mode=="min"){
        chmin(D[k],D[j]);
      }
      else if(mode=="max"){
        chmax(D[k],D[j]);
      }
      else if(mode=="sum"){
        D[k] += D[j];
      }
      j=k;
    }    
  }
  void print(){
    dump(D);
    REPM(i,2*N,1){
      cerr << D[i] << " ";
      if(__builtin_popcountll(i+1)==1){
        cerr << endl;
      }
    }
  }
  LL query(LL a ,LL b, LL k=1, LL l=0 ,LL r=-1){
    if(r==-1){
      r = N;
    }
    if(b<=l || r <= a ){ return DEFAULT;}
    if(a<=l && r <= b ){ return D[k];}
    LL vl = query(a,b, 2*k  , l ,(l+r)/2);
    LL vr = query(a,b, 2*k+1, (l+r)/2,r);
    //dump(vl,vr);
    if(mode=="min"){
      return min({vl,vr});    
    }
    else if(mode=="max"){
      return max({vl,vr});    
    }
    else if(mode=="sum"){
      return vl+vr;    
    }
  }
};

void oira_tua(LL me, LL oya,VVLL &T,VLL &IO){
  IO.push_back(me);
  dump(me,oya);
  for(LL i=0;i<(LL)T[me].size();i++){
    LL ko = T[me][i];
    if(ko==oya){
      continue;
    }
    oira_tua(ko,me, T,IO);
    IO.push_back(me);
  }
}
VVLL cinvll2(LL H,LL W,LL bias = 0){
  VVLL A = zerosll(H,W);
  REP(i,H){
    REP(j,W){
      cin >> A[i][j];
      A[i][j] += bias;
    }
  }
  return move(A);
}    

void print(vector<priority_queue<LL>> P){

  REP(i,P.size()){
    auto p = P[i];
    cerr <<i << " : ";
    while(!p.empty()){
      LL f = p.top(); p.pop();
      cerr << f <<" ";
    }
    cerr << endl;
  }
} 
void print(vector<multiset<LL,greater<LL>>> P){

  REP(i,P.size()){
    auto p = P[i];
    cerr <<i << " : ";
    for(auto f: p){
      cerr << f <<" ";
    }
    cerr << endl;
  }
} 

void print(vector<map<LL,LL,greater<LL>>> P){

  REP(i,P.size()){
    auto p = P[i];
    cerr <<i << " : ";
    for(auto f: p){
      cerr << f <<" ";
    }
    cerr << endl;
  }
} 


tuple<LD,LD,LD> point2line(LD y1,LD x1,LD y2,LD x2){
  LD A = y1-y2;
  LD B = x2-x1;
  LD C = -A*x1-B*y1;
  return make_tuple(A,B,C);
}
LD dist_point_line(LD y1,LD x1,LD A,LD B,LD C){
  return abs(A*x1+B*y1+C)/sqrtl(A*A+B*B);
}

string rec(LL n,LL k, VLL &C){
  if(k==0){
    if(n==0){
      return "A";
    }
    if(n==1){
      return "B";
    }    
    return "C";
  }

  if(n==0){
    return "A";
  }
  n -=1;
  if(n>=0 and n<C[k-1]){
    return rec(n,k-1,C);
  }
  n -= C[k-1];
  if(n==0){
    return "B";
  }
  n -=1;
  if(n>=0 and n<C[k-1]){
    return rec(n,k-1,C);
  }
  n -= C[k-1];
  if(n==0){
    return "C";
  }
  dump("error");
  return "-";
}


VLL permute(VLL src, VLL idx){
  VLL dst(src.size());
  REP(i,src.size()){
    dst[i] = src[idx[i]];
  }
  return move(dst);
}


bool isOK(LL b,VLL &Futosa, VLL &Price,LL K,LL N,LL Yosan){
  auto it = lower_bound(ALL(Futosa),b);
  LL index = distance(Futosa.begin(),it);
  dump(b,index);
  dump(Futosa);
  if(N-index<K){
    dump("tariai",N,index,K);
    return false;
  }

  VLL Z;
  REPM(i,N,index){
    Z.push_back(Price[i]);
  }
  sort(ALL(Z));
  LL cnt=0;
  REP(i,K){
    cnt += Z[i];
  }
  if(cnt <= Yosan){
   dump("OK",N,b,index,K,cnt,Yosan);
    return true;
  }
  dump("over",N,index,K,cnt,Yosan);
  return false;
}



void dfs(LL me, LL oya, VVVLL &T,VLL &C){

  for(auto t : T[me]){
    LL ko = t[0];
    LL w = t[1];
    if(ko==oya){continue;}
    if(w%2==0){
      C[ko] = C[me];
      dump(me, ":", ko,w,"same");
    }
    else{
      C[ko] = 1-C[me];
      dump(me, ":", ko,w,"diff");
    }
    dfs(ko,me,T,C);    
  }
}


tuple<VLD,VLL,VLL> my_dycstra_v1(vector<vector<pair<LL,LD>>> &R,LL start, LL MOD = 0)
{
  LL N = R.size();
  LL INFINF = 1e15;
  using PQ = pair<LD,LL>;
  priority_queue<PQ, vector<PQ>, greater<PQ>> Q;
  VLD Cost(N, INFINF);
  VLL Path(N, -1);
  VLL Num(N, 0);
  Cost[start] = 0;
  Num[start] = 1;
  Q.push({0,start});
  LL cnt=0;
  while(!Q.empty()) {
    //dump(Q.size());
    PQ q = Q.top();
    Q.pop();
    LD cost = q.first;
    LL from = q.second;

    if (Cost[from] < cost){continue;}

    //dump(from,cost);
    for(auto r : R[from]) {
      cnt++;
      LL to = r.first;
      LD cost_to = r.second;
      //dump(to,cost_to);
      
      if(chmin(Cost[to], (cost + cost_to))) {
        Q.push({cost + cost_to, to});
        Path[to] = from;
        Num[to] = Num[from];
      }
      else if(Cost[to] == (cost + cost_to)){
        Num[to] += Num[from];
        if(MOD!=0){
          Num[to] %= MOD;
        }
      }
    }
  }
  dump(cnt);
  // dump(C);
  return move(make_tuple(Cost,Path,Num));
}

void func()
{
  LL Xs,Ys,Xt,Yt;
  cin >> Xs>>Ys>>Xt>>Yt;
  LL N ;
  cin >> N;
  VVLL B = cinvll2(N,3);
  VVLL A = zerosll(N+2,3);
  A[0] = {Xs,Ys,0};
  A[1] = {Xt,Yt,0};
  REP(i,N){
    A[i+2] = B[i];
  }
  //dump(A);
  dump("start0");
  N +=2;
  UnionFindTree uft(N);
  VVLD D(N,VLD(N,0));
  REP(i,N-1){
    LD x0 = A[i][0];
    LD y0 = A[i][1];
    LD r0 = A[i][2];
    REPM(j,N,i+1){
      LD x1 = A[j][0];
      LD y1 = A[j][1];
      LD r1 = A[j][2];
      LD dist = sqrtl((x0-x1)*(x0-x1)+(y0-y1)*(y0-y1));
      dist -=(r0+r1);
      chmax(dist,(LD)0.0);
      D[i][j] = dist;
      D[j][i] = dist;
      if(dist==0.0){
        uft.unite(i,j);
      }
    }
  }  
  //dump(D);
  dump("start1");

  //uft.print();
  LL M = uft.get_root_num();
  unordered_map<LL, vector<LL>> U = uft.get_root_child2();
  VVLL V;
  map<LL,LL> T;
  for(auto u : U){
    T[u.first] = V.size();
    V.push_back(u.second);
  }
  //dump(T);
  //(V);
  vector<vector<pair<LL,LD>>> R(M);
  dump("start2");

  REP(i,M-1){
    REPM(j,M,i+1){
      LD dist = 1e15;
      for(auto ui : V[i]){
        for(auto uj : V[j]){
          //dump(ui,uj,D[ui][uj]);
          chmin(dist,D[ui][uj]);
        }
      }
      R[i].push_back({j,dist});
      R[j].push_back({i,dist});
     // dump(i,j,dist);
    }
  }
    dump("start3");

  //dump(R);
  //uft.print();
  //dump(D);
  VLD Cost;
  VLL Path,Num;
  
  tie(Cost,Path,Num) = my_dycstra_v1(R,T[uft.root(0)]);
  //dump(Cost);
  bye(Cost[T[uft.root(1)]]);


/*
  A.push_back({Xs,Ys,0});
  A.push_back({Xt,Yt,0});
  N = A.size();
  vector<vector<pair<LL,LD>>> R(N);
  REP(i,N-1){
    LD x0 = A[i][0];
    LD y0 = A[i][1];
    LD r0 = A[i][2];
    REPM(j,N,i+1){
      LD x1 = A[j][0];
      LD y1 = A[j][1];
      LD r1 = A[j][2];

      LD dist = sqrtl((x0-x1)*(x0-x1)+(y0-y1)*(y0-y1));
      dist -=(r0+r1);
      chmax(dist,(LD)0.0);
      R[i].push_back({j,dist});
      R[j].push_back({i,dist});
    }
  }
  VLD Cost;
  VLL Path,Num;
  tie(Cost,Path,Num) = my_dycstra_v1(R,N-2);
  dump(Cost);
  bye(Cost[N-1]);
  */
}
int main()
{
  func();
  return 0;
}



Measured memory_bytes: 58220