#include <bits/stdc++.h>
using namespace std;

void chmin(int& a, int b){
    a = min(a, b);
}
int nth_bit(int64_t num, int n){
    return (num >> n) & 1;
}

int main(){
    int N;
    int A[100000] = {0};
    cin >> N;
    for(int i=0; i<N-1; i++){
        int x, y, a;
        cin >> x >> y >> a;
        A[x] ^= a;
        A[y] ^= a;
    }

    int num[16] = {0};
    for(int i=0; i<N; i++) num[A[i]]++;

    int fix = 0;
    vector<int> rest;
    for(int i=1; i<16; i++){
        fix += num[i] / 2;
        if(num[i] % 2) rest.push_back(i);
    }

    int sz = rest.size();
    int M = 1<<sz;
    const int INF = 1e9;
    static int dp[1<<15];
    dp[0] = fix;
    for(int i=1; i<M; i++) dp[i] = INF;

    for(int i=1; i<M; i++){
        for(int j=i; j>0; j=(j-1)&i){
            int s = 0, c = 0;
            for(int k=0; k<sz; k++) if(nth_bit(j, k)){
                c++;
                s ^= rest[k];
            }
            if(s == 0) chmin(dp[i], dp[i-j] + c-1);
        }
    }

    cout << dp[M-1] << endl;
    return 0;
}

### Prompt

Your task is to create a CPP solution to the following problem:

You are given a tree with N vertices. The vertices are numbered 0 through N-1, and the edges are numbered 1 through N-1. Edge i connects Vertex x_i and y_i, and has a value a_i. You can perform the following operation any number of times:

* Choose a simple path and a non-negative integer x, then for each edge e that belongs to the path, change a_e by executing a_e ← a_e ⊕ x (⊕ denotes XOR).



Your objective is to have a_e = 0 for all edges e. Find the minimum number of operations required to achieve it.

Constraints

* 2 ≤ N ≤ 10^5
* 0 ≤ x_i,y_i ≤ N-1
* 0 ≤ a_i ≤ 15
* The given graph is a tree.
* All input values are integers.

Input

Input is given from Standard Input in the following format:


N
x_1 y_1 a_1
x_2 y_2 a_2
:
x_{N-1} y_{N-1} a_{N-1}


Output

Find the minimum number of operations required to achieve the objective.

Examples

Input

5
0 1 1
0 2 3
0 3 6
3 4 4


Output

3


Input

2
1 0 0


Output

0

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;

void chmin(int& a, int b){
    a = min(a, b);
}
int nth_bit(int64_t num, int n){
    return (num >> n) & 1;
}

int main(){
    int N;
    int A[100000] = {0};
    cin >> N;
    for(int i=0; i<N-1; i++){
        int x, y, a;
        cin >> x >> y >> a;
        A[x] ^= a;
        A[y] ^= a;
    }

    int num[16] = {0};
    for(int i=0; i<N; i++) num[A[i]]++;

    int fix = 0;
    vector<int> rest;
    for(int i=1; i<16; i++){
        fix += num[i] / 2;
        if(num[i] % 2) rest.push_back(i);
    }

    int sz = rest.size();
    int M = 1<<sz;
    const int INF = 1e9;
    static int dp[1<<15];
    dp[0] = fix;
    for(int i=1; i<M; i++) dp[i] = INF;

    for(int i=1; i<M; i++){
        for(int j=i; j>0; j=(j-1)&i){
            int s = 0, c = 0;
            for(int k=0; k<sz; k++) if(nth_bit(j, k)){
                c++;
                s ^= rest[k];
            }
            if(s == 0) chmin(dp[i], dp[i-j] + c-1);
        }
    }

    cout << dp[M-1] << endl;
    return 0;
}

```