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<iostream>
#include<cstdio>
#include<cstdlib>
#include<algorithm>
#include<vector>
#include<string>
#include<sstream>
#include<iomanip>
#include<utility>
#include<cmath>
#include<set>
#include<list>
#include<string>
#include<queue>
#include<stack>
#include<deque>
#include<map>
#include<set>
#include<cstring>
#include<iterator>
#include<bitset>
#include<assert.h>
//#include<unistd.h>
//#include<chrono>
//#include<time.h>
//#include<windows.h>
using namespace std;
 
//#define int ll
typedef long long ll;
static const int MOD = 1000000007;
//static const int INF = 2147483647;
//static const long long INF = 9223372000000000000;
//static const long long INF = 9223372000000000000/2;
//static const int INF = 1000010000;
//int dx4[4] = {0,1,0,-1}, dy4[4] = {-1,0,1,0};
//int dx5[5] = {-1,0,0,0,1}, dy5[5] = {0,-1,0,1,0};
//int dx8[8] = {-1,0,1,1,1,0,-1,-1}, dy8[8] = {1,1,1,0,-1,-1,-1,0};
//int dx9[9] = {-1,0,1,1,1,0,-1,-1,0}, dy9[9] = {1,1,1,0,-1,-1,-1,0,0};
#define PB push_back
#define EB emplace_back
#define MP make_pair
#define MT make_tuple
#define FI first
#define SE second
#define NP next_permutation
#define PQ priority_queue
#define UB upper_bound
#define LB lower_bound
#define SZ(a) int((a).size())
#define LEN(a) int((a).length())
#define SORT(c) sort((c).begin(),(c).end())
#define FOR(i,a,b) for(int i=(a);i<(b);i++)
#define REP(i,x) for(int i=0;i<(int)(x);i++)
#define REP1(i,x) for(int i=1;i<=(int)(x);i++)
#define RREP(i,x) for(int i=((int)(x)-1);i>=0;i--)
#define RREP1(i,x) for(int i=((int)(x));i>0;i--)
#define ALL(x) (x).begin(),(x).end()
#define YESNO(x) puts((x)?"YES":"NO")

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) {}
 
  operator int() const { return to; }
};

template< typename T >
using Edges = vector< edge< T > >;
template< typename T >
using WeightedGraph = vector< Edges< T > >;

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;
}


signed main(){
    int sx,sy,gx,gy;
    cin >> sx >> sy >> gx >> gy;
    int n;
    cin >> n;
    pair<pair<int,int>,int> z[1145];
    z[0] = MP(MP(sx,sy),0);
    z[n+1] = MP(MP(gx,gy),0);

    REP1(i,n){
        int x,y,r;
        cin >> x >> y >> r;
        z[i] = MP(MP(x,y),r);
    }

    WeightedGraph< double > g(n+2);
    REP(i,n+2){
        FOR(j,i+1,n+2){
            double dist = 
                hypot(double(abs(z[i].FI.FI-z[j].FI.FI)),
                      double(abs(z[i].FI.SE-z[j].FI.SE)))
                - double(z[i].SE+z[j].SE);
            if(dist<0) dist = 0;
            g[i].EB(j,dist);
            g[j].EB(i,dist);
        }
    }

    auto cost = dijkstra(g,0);
    printf("%.15lf\n",cost[n+1]);

    return 0;
}