E - Xor Distances

Score : $500$ points

Problem Statement

We have a weighted tree with $N$ vertices. The $i$-th edge connects Vertex $u_i$ and Vertex $v_i$ bidirectionally and has a weight $w_i$.

For a pair of vertices $(x,y)$, let us define $\text{dist}(x,y)$ as follows:

• the XOR of the weights of the edges in the shortest path from $x$ to $y$.

Find $\text{dist}(i,j)$ for every pair $(i,j)$ such that $1 \leq i \lt j \leq N$, and print the sum of those values modulo $(10^9+7)$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq u_i \lt v_i \leq N$
• $0 \leq w_i \lt 2^{60}$
• The given graph is a tree.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$u_1$ $v_1$ $w_1$

$u_2$ $v_2$ $w_2$

$\vdots$

$u_{N-1}$ $v_{N-1}$ $w_{N-1}$

Output

Print the sum of $\text{dist}(i,j)$, modulo $(10^9+7)$.

3
1 2 1
1 3 3

6

We have $\text{dist}(1,2)=1,$ $\text{dist}(1,3)=3,$ and $\text{dist}(2,3)=2$, for the sum of $6$.

5
3 5 2
2 3 2
1 5 1
4 5 13

62

10
5 7 459221860242673109
6 8 248001948488076933
3 5 371922579800289138
2 5 773108338386747788
6 10 181747352791505823
1 3 803225386673329326
7 8 139939802736535485
9 10 657980865814127926
2 4 146378247587539124

241240228

Print the sum modulo $(10^9+7)$.