Problem:
Score : 600 points

Problem Statement

On the xy-plane, Snuke is going to travel from the point (x_s, y_s) to the point (x_t, y_t).
He can move in arbitrary directions with speed 1.
Here, we will consider him as a point without size.

There are N circular barriers deployed on the plane.
The center and the radius of the i-th barrier are (x_i, y_i) and r_i, respectively.
The barriers may overlap or contain each other.

A point on the plane is exposed to cosmic rays if the point is not within any of the barriers.

Snuke wants to avoid exposure to cosmic rays as much as possible during the travel.
Find the minimum possible duration of time he is exposed to cosmic rays during the travel.

Constraints

All input values are integers.

-10^9 ≤ x_s, y_s, x_t, y_t ≤ 10^9

(x_s, y_s) ≠ (x_t, y_t)

1≤N≤1,000

-10^9 ≤ x_i, y_i ≤ 10^9

1 ≤ r_i ≤ 10^9

Input

The input is given from Standard Input in the following format:

x_s y_s x_t y_t
N
x_1 y_1 r_1
x_2 y_2 r_2
:
x_N y_N r_N

Output

Print the minimum possible duration of time Snuke is exposed to cosmic rays during the travel.
The output is considered correct if the absolute or relative error is at most 10^{-9}.

Sample Input 1

-2 -2 2 2
1
0 0 1

Sample Output 1

3.6568542495

An optimal route is as follows:

Sample Input 2

-2 0 2 0
2
-1 0 2
1 0 2

Sample Output 2

0.0000000000

An optimal route is as follows:

Sample Input 3

4 -2 -2 4
3
0 0 2
4 0 1
0 4 1

Sample Output 3

4.0000000000

An optimal route is as follows:

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>

using namespace std;

#define pb push_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second
#define all(a) (a).begin(),(a).end()
#define endl "\n"
#define rep(i, a, b) for (auto (i) = (a); (i) < (b); (i)++)
#define rrep(i, a, b) for (auto (i) = (a); (i) > (b); (i)--)

#define aim_cpo do{ios::sync_with_stdio(false);\
cin.tie(nullptr);cout.precision(12);cout<<fixed;}while(0)

#define LOCAL_INPUT do{FILE *stream1;\
stream1=freopen("/Users/aim_cpo/CLionProjects/competitive/in.txt","r",stdin);\
if(stream1==nullptr)return 0;}while(0)
#define LOCAL_OUTPUT do{FILE *stream2;\
stream2 = freopen("out.txt", "w", stdout);\
if (stream2 == nullptr) return 0;}while(0)

#ifdef LOCAL_DEFINE
#define show(x) cerr << #x << " = " << (x) << endl
#define dump(x) cerr << #x << " = " << (x) << " (line " << __LINE__ << ")" << endl
#define showV(v, n) do{\
for(int _i_t_e_=0; _i_t_e_<(n);_i_t_e_++)\
cerr<<"("<<_i_t_e_<<" = "<<(v)[_i_t_e_]<<") ";\
cerr<<endl;}while(0)
#define showA(a, n) do{\
for(int _i_t_e_=0;_i_t_e_<(n);_i_t_e_++)\
cerr<<"("<<_i_t_e_<<" = "<<(a)[_i_t_e_]<<") ";\
cerr<<endl;}while(0)
#define showA2(a, n, m) do{\
for(int _i_t_e_=0;_i_t_e_<(n);_i_t_e_++){\
for(int _i_t_e_2=0;_i_t_e_2<(m);_i_t_e_2++){\
cerr<<"("<<_i_t_e_<<", "<<_i_t_e_2<<" = "<<(a)[_i_t_e_][_i_t_e_2]<<") ";\
}cerr<<endl;}cerr<<endl;}while(0)
#else
#define show(x)
#define dump(x)
#define showV(v, n)
#define showA(a, n)
#define showA2(a, n, m)
#endif

typedef long long ll;
typedef unsigned long long ull;

constexpr const int INT_INF=0x3f3f3f3f; //1061109567
constexpr const ll LL_INF=0x3f3f3f3f3f3f3f3f; //4557430888798830399

template <typename T> bool chmin(T &a, T b){return a>b?(a=b,true):false;}
template <typename T> bool chmax(T &a, T b){return a<b?(a=b,true):false;}
template <typename T> void ln(T i, T n){cout<<(i==n-1?"\n":" ");}
template <typename T, typename S>
ostream &operator<<(ostream &out,const pair<T, S> &pair1){
  out<<'('<<pair1.fi<<", "<<pair1.se<<')';return out;}
template <typename T>
ostream &operator<<(ostream &out,const vector<T> &vector1){
  out<<'[';for(unsigned int i=0;i<vector1.size();i++){
    out<<vector1[i];if(i==vector1.size()-1)out<<"]";else out<<", ";
  }return out;
}

// INT
#define GCD(a, b) __gcd(a, b)
template <typename T> T LCM(T a, T b) {return a / GCD(a, b) * b;}
template <typename T> T EXTGCD(T a, T b, T& x, T& y) {
  T d = a;
  if (b != 0) {d=EXTGCD(b,a%b,y,x);y-=(a/b)*x;}
  else x=1,y=0;
  return d;
}
template <typename T> bool is_prime(T a) {
  for(int i=2;i*i<=a;i++)if(a%i==0)return 1;
  return 0;
}
template <typename T, typename S> T Pow(T a, S b) {
  T res = 1, now = a;
  while(b){if(b&1)res*=now;b>>=1;now*=now;}return res;
}

// MOD
ll MOD = 1000000000L + 7L;
#define add(a, b) ((a % MOD) + (b % MOD)) % MOD
#define mul(a, b) ((a % MOD) * (b % MOD)) % MOD
#define sub(a, b) ((a % MOD) + MOD - (b % MOD)) % MOD
template <typename T, typename S> T ModPow(T a, S b) {
  T res = 1, now = a;
  while(b){if(b&1)res=mul(res,now);b>>=1;now=mul(now,now);}return res;
}
template <typename T> T mod_inverse(T a, T mod, bool prime){ // if mod is prime, "prime" is true.
  if(prime)return ModPow(a, mod - 2);
  else{T x,y;EXTGCD(a,mod,x,y);return (mod+x%mod)%mod;}
}
#define divide(a, b) ((a % MOD) * (mod_inverse(b, MOD, true))) % MOD

//LLの数値をつかう時は最後にLをつける癖をつけよう
//
//  ┏┓┏┓　┓　┏┓
//  ┏┛┃┃　┃　┗┫
//  ┗┛┗┛　┻　┗┛
//
//   謹┃賀┃新┃年┃
//   ━┛━┛━┛━┛
//WWWWWWWWWWWWWWWWWWWWWW

vector < pair < pair <int, int>, int> > v;
vector < pair <int, double> > g[1011];
double d[1011];

int main() {
  aim_cpo; // インタラクティブのときは消すように.
#ifdef LOCAL_DEFINE
  LOCAL_INPUT; // インタラクティブのときは消すように.
//  LOCAL_OUTPUT; // ファイルに出力したいときのみ
  show(MOD);
#endif
  // テキトー考察
  //
  // いろんな行き方がありそうだから、ダイクストラかな...(でもどうやって辺を貼れば良いんだろう)
  // 例えば、ある座標からある座標に行くときに、その途中で関係ない場所のバリアを通ってしまうようなときは辺のコストをどんな値にすれば
  // 良いのだろう...
  // でも、よく考えたらそんな経路が最短になるわけないよね.
  // 途中で、どこかのバリアをかすってしまうなら、そのバリアの中心に向かって移動して、そこからまた目的の場所に行くほうが
  // 絶対に短くなるはず
  // じゃあ、どうせその経路は使われないんだから、厳密なコストを求める必要はなくて、それよりもう少し長いコストでも良さそう.
  // じゃあ、シンプルに座標同士の距離をコストにすれば丁度いい感じに求められそう.
  // 個人的に、辺のコストを求める部分がこの問題の本質だなって思った.
  // ある座標からある座標への経路の中に、絶対使われない経路が存在することに気づくことが大切.
  int sx, sy; cin >> sx >> sy;
  int gx, gy; cin >> gx >> gy;
  v.push_back(mp(mp(sx, sy), 0));
  v.push_back(mp(mp(gx, gy), 0));
  int n; cin >> n;
  rep(i, 0, n) {
    int x, y, r; cin >> x >> y >> r;
    v.push_back(mp(mp(x, y), r));
  }
  rep(i, 0, (int)v.size()) {
    rep(j, 0, (int)v.size()) {
      if (i == j) continue;
      double cost = sqrt(Pow((double)v[i].fi.fi - v[j].fi.fi, 2) + Pow((double)v[i].fi.se - v[j].fi.se, 2));
      if (cost - v[i].se - v[j].se < 0) cost = 0.;
      else cost -= v[i].se + v[j].se;
      g[i].emplace_back(j, cost);
    }
  }
  priority_queue<pair<double, int> > q;
  fill(d, d + n + 2, 1e17);
  d[0] = 0.;
  q.push(mp(0., 0));
  while (!q.empty()) {
    int now = q.top().se;
    double cost = -q.top().fi; q.pop();
    if (d[now] < cost) continue;
    rep(i, 0, (int)g[now].size()) {
      if (d[g[now][i].fi] > d[now] + g[now][i].se) {
        d[g[now][i].fi] = d[now] + g[now][i].se;
        q.push(mp(-d[g[now][i].fi], g[now][i].fi));
      }
    }
  }
  cout << d[1] << endl;
#ifdef  LOCAL_DEFINE
  cerr << "Time elapsed: " << 1.0 * clock() / CLOCKS_PER_SEC << "s." << endl;
#endif
  return 0;
}