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Contest: Task: Related: TaskD TaskF

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 $(10^9+7)$.

What is $\text{ XOR }$?

The bitwise $\mathrm{XOR}$ of integers $A$ and $B$, $A\ \mathrm{XOR}\ B$, is defined as follows:

When $A\ \mathrm{XOR}\ B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of $A$ and $B$ is $1$, and $0$ otherwise.

For example, we have $3\ \mathrm{XOR}\ 5 = 6$ (in base two: $011\ \mathrm{XOR}\ 101 = 110$).

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 $(10^9+7)$.

Sample Input 1

3
1 2 1
1 3 3

Sample Output 1

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

Sample Input 2

Sample Output 2

Sample Input 3

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

Sample Output 3

241240228

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