Problem: problem

AOR Ika likes rooted trees with a fractal (self-similar) structure. Consider using the weighted rooted tree $ T $ consisting of $ N $ vertices to represent the rooted tree $ T'$ with the following fractal structure.

* $ T'$ is the addition of a tree rooted at $ x $ and having a tree structure similar to $ T $ (same cost) for each vertex $ x $ of $ T $.
* The root of $ T'$ is the same as that of $ T $.



The tree represented in this way is, for example, as shown in the figure below.

<image>

AOR Ika is trying to do a depth-first search for $ T'$, but finds that it takes a lot of time to trace all the vertices. Therefore, we decided to skip some node visits by performing a depth-first search with a policy of transitioning with a probability of $ p $ and not transitioning with a probability of $ 1-p $ at the time of transition during a depth-first search. Given $ T $ and the probability $ p $, find the expected value of the sum of the costs of all edges to follow when performing a depth-first search for $ T'$. The information of $ T $ is given by the number of vertices $ N $ and the information of the edges of $ N-1 $, and the vertex $ 1 $ is the root. Each vertex is labeled $ 1,2, \ dots, N $, and the $ i \ (1 \ le i \ le N-1) $ th edge costs the vertices $ x_i $ and $ y_i $ $ c_i $ It is tied with. The non-deterministic algorithm of the depth-first search that transitions to a child with a probability of $ p $, which is dealt with in this problem, is expressed as follows. The sum of the costs of the edges that the output $ \ mathrm {answer} $ follows.

1. Prepare an empty stack $ S $.
2. Set $ ​​\ mathrm {answer} = 0 $
3. Push the root vertex of $ T'$ to $ S $.
4. Extract the first element of $ S $ and make it $ x $.
5. For each child $ c $ of $ x $, do the following with probability $ p $ and do nothing with probability $ 1-p $.
* Add vertex $ c $ to $ S $. Then add the weight of the edge connecting $ x $ to $ c $ to $ \ mathrm {answer} $.
6. If S is not empty, transition to 3.
7. Output $ \ mathrm {answer} $.



output

Print the answer in one line. If the relative or absolute error is less than $ 10 ^ {-6} $, then AC. Also, output a line break at the end.

Example

Input

0.75
4
1 2 1
2 3 3
3 4 10


Output

24.8569335938

Solution (in CPP):

Solution:
#include<cstdio>
#include<vector>
#include<math.h>

using namespace std;

const int MAX_N = 100001;

struct edge {int to,cost;};
vector<edge> tree[MAX_N];
bool flag[MAX_N] = {false};
int depth[MAX_N];
int s,t,c;
double sum = 0.0;
double ans = 0.0;
double p;
int N;

void dfs(int v,int d){
	flag[v] = true;
	depth[v] = d;
	for(int i = 0;i < tree[v].size();i++){
		if(!flag[tree[v][i].to]){
			sum += tree[v][i].cost * pow(p,(d+1.0));
			dfs(tree[v][i].to,d+1);
		}
	}
}

int main(){
	scanf("%lf %d",&p,&N);
	
	for(int i =0; i<N-1; i++){
		scanf("%d %d %d",&s,&t,&c);
		s--;
		t--;
		edge e;
		e.to = t;
		e.cost = c;
		tree[s].push_back(e);
		e.to = s;
		tree[t].push_back(e);
	}
	
	dfs(0,0);
	
	ans = 2*sum;
	for(int i = 1; i < N;i++){
		ans += sum * pow(p,depth[i]);
	}
	printf("%.10lf\n",ans);
	
	return 0;
}