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 <bits/stdc++.h>
using namespace std;
using P = pair<int, int>;

vector<bool> used;
vector<vector<P>> g;
vector<P> num;

void dfs(int s, int dep) {
    for (auto&& p : g[s]) {
        int v, c; tie(v, c) = p;
        if (used[v]) continue;
        used[v] = true;
        num[v] = {dep + 1, c};
        dfs(v, dep + 1);
    }
}

int main() {
    double p; cin >> p;
    int N; cin >> N;
    g = vector<vector<P>>(N);
    for (int i = 0; i < N-1; ++i) {
        int x, y, c; cin >> x >> y >> c; --x; --y;
        g[x].push_back({y, c});
        g[y].push_back({x, c});
    }
    used = vector<bool>(N, false);
    num = vector<P>(N, {0, 0});
    used[0] = true;
    dfs(0, 0);

    double t = 0;
    for (int i = 1; i < N; ++i) t += pow(p, num[i].first) * num[i].second;
    double d = 0;
    for (int i = 0; i < N; ++i) d += pow(p, num[i].first);

    cout << fixed << setprecision(7) << t + t * d << endl; 
}