Problem:
Score : 700 points

Problem Statement

We have a graph with N vertices and M edges, and there are two people on the graph: Takahashi and Aoki.

The i-th edge connects Vertex U_i and Vertex V_i.
The time it takes to traverse this edge is D_i minutes, regardless of direction and who traverses the edge (Takahashi or Aoki).

Takahashi departs Vertex S and Aoki departs Vertex T at the same time. Takahashi travels to Vertex T and Aoki travels to Vertex S, both in the shortest time possible.
Find the number of the pairs of ways for Takahashi and Aoki to choose their shortest paths such that they never meet (at a vertex or on an edge) during the travel, modulo 10^9 + 7.

Constraints

1 \leq N \leq 100 000

1 \leq M \leq 200 000

1 \leq S, T \leq N

S \neq T

1 \leq U_i, V_i \leq N (1 \leq i \leq M)

1 \leq D_i \leq 10^9 (1 \leq i \leq M)

If i \neq j, then (U_i, V_i) \neq (U_j, V_j) and (U_i, V_i) \neq (V_j, U_j).

U_i \neq V_i (1 \leq i \leq M)

D_i are integers.

The given graph is connected.

Input

Input is given from Standard Input in the following format:

N M
S T
U_1 V_1 D_1
U_2 V_2 D_2
:
U_M V_M D_M

Output

Print the answer.

Sample Input 1

4 4
1 3
1 2 1
2 3 1
3 4 1
4 1 1

Sample Output 1

2

There are two ways to choose shortest paths that satisfies the condition:

Takahashi chooses the path 1 \rightarrow 2 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 4 \rightarrow 1.

Takahashi chooses the path 1 \rightarrow 4 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 2 \rightarrow 1.

Sample Input 2

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 2

2

Sample Input 3

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 3

2

Sample Input 4

8 13
4 2
7 3 9
6 2 3
1 6 4
7 6 9
3 8 9
1 2 2
2 8 12
8 6 9
2 5 5
4 2 18
5 3 7
5 1 515371567
4 8 6

Sample Output 4

6

Solution in C (GCC 5.4.1) (judge verdict: Wrong Answer)

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

int max(int a,int b){if(a>b){return a;}return b;}
int min(int a,int b){if(a<b){return a;}return b;}
int zt(int a,int b){return max(a,b)-min(a,b);}
int round(int a,int b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
int ceil(int a,int b){if(a%b==0){return a/b;}return (a/b)+1;}
int gcd(int a,int b){int c;while(b!=0){c=a%b;a=b;b=c;}return a;}
int lcm(int a,int b){int c=gcd(a,b);a/=c;return a*b;}
int nCr(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
int nHr(int a,int b){return nCr(a+b-1,b);}
int fact(int a){int i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
int pow(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
int dsum(int x){int r=0;while(x){r+=(x%10);x/=10;}return r;}
int dsumb(int x,int b){int r=0;while(x){r+=(x%b);x/=b;}return r;}
int sankaku(int x){return ((1+x)*x)/2;}
long long llmax(long long a,long long b){if(a>b){return a;}return b;}
long long llmin(long long a,long long b){if(a<b){return a;}return b;}
long long llzt(long long a,long long b){return llmax(a,b)-llmin(a,b);}
long long llround(long long a,long long b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
long long llceil(long long a,long long b){if(a%b==0){return a/b;}return (a/b)+1;}
long long llgcd(long long a,long long b){long long c;while(b!=0){c=a%b;a=b;b=c;}return a;}
long long lllcm(long long a,long long b){long long c=llgcd(a,b);a/=c;return a*b;}
long long llnCr(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
long long llnHr(long long a,long long b){return llnCr(a+b-1,b);}
long long llfact(long long a){long long i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
long long llpow(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
long long lldsum(long long x){long long r=0;while(x){r+=(x%10);x/=10;}return r;}
long long lldsumb(long long x,long long b){long long r=0;while(x){r+=(x%b);x/=b;}return r;}
long long llsankaku(long long x){return ((1+x)*x)/2;}
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 sortfncsj(const void *a,const void *b){if(*(int *)a>*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
int sortfnckj(const void *a,const void *b){if(*(int *)a<*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
int llsortfncsj(const void *a,const void *b){if(*(long long *)a>*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
int llsortfnckj(const void *a,const void *b){if(*(long long *)a<*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
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;}
int strsortfncsj(const void *a,const void *b){return strcmp((char *)a,(char *)b);}
int strsortfnckj(const void *a,const void *b){return strcmp((char *)b,(char *)a);}

//Dynamic Array
 
typedef struct{
    long long st;
    long long fi;
    long long 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;
}
 
//Heap and Priority_queue
 
typedef struct{
long long node;
long long cost;
long long from;
}data;
 
data heap[1048576];
long long hsize=0;
 
void resheap(){
    long long i;
    heap[0].node=-1;
    heap[0].cost=-llinf;
    for(i=1;i<1048576;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;
}

int main(void){
    resheap();
    long long i,j,n,m,s,t,u,v,d,p,q,sd,avalon;
    long long sst,sft,tst,tft;
    long long sw=0,tw=0,dw=0;
    long long ss[524288],ts[524288];
    long long sp[524288]={0},tp[524288]={0};
    bool sef[524288]={0},tef[524288]={0};
    data pd,od;
    rs g[524288];
    mkj x[524288];
    scanf("%lld%lld%lld%lld",&n,&m,&s,&t);
    for(i=0;i<m;i++){
      scanf("%lld%lld%lld",&u,&v,&d);
      g[2*i].st=u;
      g[2*i].fi=v;
      g[2*i].kr=d;
      g[2*i+1].st=v;
      g[2*i+1].fi=u;
      g[2*i+1].kr=d;
    }
    for(i=0;i<=n;i++){
      ss[i]=llinf;
      ts[i]=llinf;
    }
    qsort(g,2*m,sizeof(g[0]),sortfnc);
    makemkj(g,x,2*m);
    pd.node=s;
    pd.cost=0;
    pd.from=-1;
    pqpush(pd);
    while(hsize>0){
      od=pqpop();
      if(ss[od.node]<od.cost){continue;}
      ss[od.node]=od.cost;
      if(od.from==-1){
        sp[od.node]++;
        sp[od.node]%=mod;
      }
      else{
        sp[od.node]+=sp[od.from];
        sp[od.node]%=mod;
      }
      for(i=x[od.node].st;i<x[od.node].st+x[od.node].kz;i++){
        if(ss[g[i].fi]!=llinf || sef[i]==1){continue;}
        sef[i]=1;
        pd.node=g[i].fi;
        pd.cost=od.cost+g[i].kr;
        pd.from=od.node;
        pqpush(pd);
      }
    }
    pd.node=t;
    pd.cost=0;
    pd.from=-1;
    pqpush(pd);
    while(hsize>0){
      od=pqpop();
      if(ts[od.node]<od.cost){continue;}
      ts[od.node]=od.cost;
      if(od.from==-1){
        tp[od.node]++;
        tp[od.node]%=mod;
      }
      else{
        tp[od.node]+=tp[od.from];
        tp[od.node]%=mod;
      }
      for(i=x[od.node].st;i<x[od.node].st+x[od.node].kz;i++){
        if(ts[g[i].fi]!=llinf || tef[i]==1){continue;}
        tef[i]=1;
        pd.node=g[i].fi;
        pd.cost=od.cost+g[i].kr;
        pd.from=od.node;
        pqpush(pd);
      }
    }
    //for(i=1;i<=n;i++){printf("<%lld> sp:%lld tp:%lld\n",i,sp[i],tp[i]);}
    sd=ss[t];
    //printf("mindist:%lld\n",sd);
    if(sd%2==0){
      for(i=1;i<=n;i++){
        if(ss[i]==(sd/2) && ts[i]==(sd/2)){
          sw+=sp[i];
          sw%=mod;
          tw+=tp[i];
          tw%=mod;
          avalon=(sp[i]*tp[i]);
          avalon%=mod;
          avalon*=avalon;
          avalon%=mod;
          dw+=avalon;
          dw%=mod;
          //printf("%lld:%lld %lld\n",i,sp[i],tp[i]);
        }
      }
    }
    for(i=0;i<2*m;i++){
      p=g[i].st;
      q=g[i].fi;
      if(sd%2==0){
        if(ss[p]==sd/2 || ss[q]==sd/2){continue;}
        if(ts[p]==sd/2 || ts[q]==sd/2){continue;}
      }
      if(ss[p]+g[i].kr+ts[q]==sd){
        sst=ss[p];
        sft=ss[q];
        tst=ts[q];
        tft=ts[p];
        if(sft<tst || tft<sst){continue;}
        sw+=sp[p];
        sw%=mod;
        tw+=tp[q];
        tw%=mod;
        dw+=(sp[p]*tp[q]);
        avalon%=mod;
        avalon*=avalon;
        avalon%=mod;
        dw+=avalon;
        dw%=mod;
        //printf("%lld %lld:[%lld %lld]\n",p,q,sp[p],tp[q]);
      }
    }
    //printf("%lld %lld\n",sp[t],tp[s]);
    printf("%lld\n",((sp[t]*tp[s])%mod+mod-dw)%mod);
    return 0;
}