You are given a tree of $n$ vertices.
You have a robot. You can place it at a vertex $u$ on the tree and order it to move to $v$ using the simple path between 2 vertices.
Moving from one vertex to another costs $1$ energy. When the robot runs out of energy, it stops.
There are $q$ queries, each of the form $(u, v, w)$, the robot starts at $u$ and moves to $v$ with $w$ energy initially. Where will it stop?
### Input
- The first line contain 2 integers $n, q$.
- The next $n - 1$ lines, each line contains 2 integers $u, v$, there is an edge between $u$ and $v$.
- The next $q$ lines, each line contains 3 integers $u, v, w$, an query.
### Output
- Print $q$ integers, the answers to $q$ queries.
### Constraints
- $1 \le n, q \le 10^5$.
- $1 \le u, v, w \le n$.
### Example
Input:
```
7 3
1 2
1 3
2 4
2 5
3 6
3 7
1 4 1
5 6 2
6 5 3
```
Output:
```
2
1
2
```
