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 (GCC 5.4.1) (judge verdict: Accepted)

//set many funcs template
//Ver.20180717
#include<stdio.h>
#include<string.h>
#include<stdlib.h>
#include<stdbool.h>
#include<time.h>
#include<math.h>
#define inf 1072114514
#define llinf 4154118101919364364
#define mod 1000000007
#define pi 3.1415926535897932384

double dbmax(double a,double b){if(a>b){return a;}return b;}
double dbmin(double a,double b){if(a<b){return a;}return b;}
double dbzt(double a,double b){return dbmax(a,b)-dbmin(a,b);}
int dbsortfncsj(const void *a,const void *b){if(*(double *)a>*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
int dbsortfnckj(const void *a,const void *b){if(*(double *)a<*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}

void shuffledget(int x[],int n){
    srand(time(0));
    int i,b[524288],p,c;
    for(i=0;i<n;i++){
        b[i]=i;
    }
    for(i=n;i>=1;i--){
        p=rand()%i;
        c=b[i-1];b[i-1]=b[p];b[p]=c;
    }
    for(i=0;i<n;i++){
        scanf("%d",&x[b[i]]);
    }
}

int dx4[4]={1,-1,0,0};
int dy4[4]={0,0,1,-1};
int dx8[8]={-1,-1,-1,0,0,1,1,1};
int dy8[8]={-1,0,1,-1,1,-1,0,1};

int search(int x,int a[],int n){
    int st=0,fi=n-1,te;
    while(st<=fi){
        te=(st+fi)/2;
        if(a[te]<x){st=te+1;}else{fi=te-1;}
    }
    return st;
}

typedef struct{
    long long st;
    long long fi;
    double kr;
}rs;
 
typedef struct{
    long long st;
    long long kz;
}mkj;
 
int sortfnc(const void *a,const void *b){
if(((rs*)a)->st == ((rs*)b)->st){return 0;}
if(((rs*)a)->st < ((rs*)b)->st){return -1;}
return 1;
}
 
void makemkj(rs g[],mkj x[],long long n){
    long long i,ms=0,nst=g[0].st;
    for(i=1;i<n;i++){
        if(g[i].st!=g[i-1].st){
            x[nst].kz=i-ms;
            x[nst].st=ms;
            nst=g[i].st;ms=i;
        }
    }
    x[nst].kz=n-ms;
    x[nst].st=ms;
}

typedef struct{
long long node;
double cost;
}data;
 
data heap[8388608];
long long hsize=0;
 
void resheap(){
    long long i;
    heap[0].node=-1;
    heap[0].cost=-llinf;
    for(i=1;i<8388608;i++){
        heap[i].node=-1;
        heap[i].cost=llinf;
    }
}
 
int swjud(data high,data low){
    if(high.cost > low.cost){return 1;}
    return -1;
}
 
void pqpush(data x){
    long long a,b;
    data m;
    hsize++;
    heap[hsize]=x;
    a=hsize;
    while(1){
        b=a/2;
        if(swjud(heap[b],heap[a]) == 1){
            m=heap[a];heap[a]=heap[b];heap[b]=m;
        }
        else{break;}
        a=b;
    }
}
 
data pqpop(){
    long long a,b,c;
    data m,sw;
    m=heap[1];
    heap[1]=heap[hsize];
    heap[hsize].node=-1;
    heap[hsize].cost=llinf;
    hsize--;
    a=1;
    while(1){
        b=a*2;c=a*2+1;
        if(swjud(heap[a],heap[b])==-1 && swjud(heap[a],heap[c])==-1){break;}
        if(swjud(heap[b],heap[c])==-1){
            sw=heap[a];heap[a]=heap[b];heap[b]=sw;
            a=b;
        }
        else{
            sw=heap[a];heap[a]=heap[c];heap[c]=sw;
            a=c;
        }
    }
    return m;
}

typedef struct{
double xz;
double yz;
double r;
}pd;

double cdist(pd a,pd b){
    double xd,yd,zd;
    xd=dbzt(a.xz,b.xz);
    yd=dbzt(a.yz,b.yz);
    zd=sqrt(xd*xd+yd*yd);
    return dbmax(0.0f,zd-a.r-b.r);
}

int main(void){
    resheap();
    data pdat,od;
    long long i,j,n,m,k,a,b,c,h,w,r=0,l,t;
    double sx,sy,fx,fy;
    double dst[1024];
    double d;
    rs g[1048576];
    mkj x[1048576];
    bool fl[1024]={0};
    pd pt[1024];
    scanf("%lf%lf",&sx,&sy);
    scanf("%lf%lf",&fx,&fy);
    scanf("%lld",&n);
    pt[0].xz=sx;
    pt[0].yz=sy;
    pt[0].r=0;
    pt[n+1].xz=fx;
    pt[n+1].yz=fy;
    pt[n+1].r=0;
    for(i=1;i<=n;i++){
        scanf("%lf%lf%lf",&pt[i].xz,&pt[i].yz,&pt[i].r);
    }
    w=0;
    for(i=0;i<=n+1;i++){
        for(j=0;j<=n+1;j++){
            if(i==j){continue;}
            g[w].st=i;
            g[w].fi=j;
            g[w].kr=cdist(pt[i],pt[j]);
            w++;
        }
    }
    qsort(g,w,sizeof(g[0]),sortfnc);
    makemkj(g,x,w);
    pdat.node=0;
    pdat.cost=0.0f;
    pqpush(pdat);
    while(hsize>0){
        od=pqpop();
        if(fl[od.node]==1){continue;}
        fl[od.node]=1;
        dst[od.node]=od.cost;
        w=od.node;
        for(i=x[w].st;i<x[w].st+x[w].kz;i++){
            if(fl[g[i].fi]==0){
                pdat.node=g[i].fi;
                pdat.cost=od.cost+g[i].kr;
                pqpush(pdat);
            }
        }
    }
    printf("%.12lf\n",dst[n+1]);
    return 0;
}