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>
 
#define _overload(_1,_2,_3,name,...) name
#define _rep(i,n) _range(i,0,n)
#define _range(i,a,b) for(int i=(int)(a);i<(int)(b);++i)
#define rep(...) _overload(__VA_ARGS__,_range,_rep,)(__VA_ARGS__)
 
#define _rrep(i,n) _rrange(i,n,0)
#define _rrange(i,a,b) for(int i=(int)(a)-1;i>=(int)(b);--i)
#define rrep(...) _overload(__VA_ARGS__,_rrange,_rrep,)(__VA_ARGS__)
 
#define _all(arg) begin(arg),end(arg)
#define uniq(arg) sort(_all(arg)),(arg).erase(unique(_all(arg)),end(arg))
#define getidx(ary,key) lower_bound(_all(ary),key)-begin(ary)
#define clr(a,b) memset((a),(b),sizeof(a))
#define bit(n) (1LL<<(n))
 
// #define DEBUG
 
#ifdef DEBUG
    #define dump(...) fprintf(stderr, __VA_ARGS__)
#else
    #define dump(...)
#endif
 
template<class T>bool chmax(T &a, const T &b) { return (a<b)?(a=b,1):0;}
template<class T>bool chmin(T &a, const T &b) { return (b<a)?(a=b,1):0;}
 
using namespace std;
using ll=long long;
using vi=vector<int>;
using vll=vector<ll>;
 
const double EPS = 1e-10;
const double PI = acos(-1.0);
const ll inf =1LL << 62;
const ll mod=1000000007LL;
const int dx[4]={1,0,-1,0};
const int dy[4]={0,1,0,-1};
 
 
ll extgcd(ll a,ll b,ll& x,ll& y){x=1,y=0;ll g=a;if(b!=0) g=extgcd(b,a%b,y,x),y-=a/b*x;return g;}
ll ADD(const ll &a, const ll &b,const ll &mod) { return (a+b)%mod;}
ll SUB(const ll &a, const ll &b,const ll &mod) { return (a-b+mod)%mod;}
ll MUL(const ll &a, const ll &b,const ll &mod) { return (1LL*a*b)%mod;}
ll DIV(const ll &a, const ll &b,const ll &mod) {ll x,y; extgcd(b,mod,x,y);return MUL(a,(x+mod)%mod,mod);}
 
random_device rd;
mt19937 mt(rd());
uniform_int_distribution<int> dice(1,6);
uniform_real_distribution<double> score(0.0,10.0);

using R = long double;
using C = tuple<R, R, R>;

inline R calc(C& ci, C& cj){
    R xi, yi, ri; tie(xi, yi, ri) = ci;
    R xj, yj, rj; tie(xj, yj, rj) = cj; 
    R ret = sqrt(pow(xi - xj, 2) + pow(yi - yj, 2)) - (ri + rj);
    return max((R)0, ret);
}

int main(void){
    cin.tie(0);
    ios::sync_with_stdio(false);

    R xs, ys, xt, yt; cin >> xs >> ys >> xt >> yt;
    int n; cin >> n;
    vector<C> in(n);
    for(auto& e : in){
        R x, y, r;
        cin >> x >> y >> r;
        e = C(x, y, r);
    }
    in.emplace_back(C(xs, ys, 0));
    in.emplace_back(C(xt, yt, 0));
    n += 2;
    int s = n - 2, t = n - 1;

    using Edge = tuple<R, int>;
    vector<vector<Edge>> G(n);
    rep(i, n){
        rep(j, i + 1, n){
            R cost = calc(in[i], in[j]);
            G[i].emplace_back(Edge(cost, j));
            G[j].emplace_back(Edge(cost, i));
        }
    }

    using Elem = tuple<R, int>;
    priority_queue<Elem, vector<Elem>, greater<Elem>> q;
    vector<R> min_dist(n, 1e10);
    q.push(Elem(0, s));
    min_dist[s] = 0;

    R res = -11;
    while(q.size()){
        R cost; int v; tie(cost, v) = q.top(); q.pop();
        if(min_dist[v] != cost) continue;
        if(v == t){
            res = cost;
            break;
        }

        for(auto& e : G[v]){
            R dcost; int nv; tie(dcost, nv) = e;
            R ncost = cost + dcost;
            if(chmin(min_dist[nv], ncost)){
                q.push(Elem(ncost, nv));
            }
        }
    }

    assert(res >= -1);
    cout.precision(20);
    cout << fixed << res << endl;

    return 0;
}