Given a tree of $n$ vertices, vertex $i$ has color $A_i$.
Given $q$ queries of the form $(u, v)$, find the dominating color on the simple path from $u$ to $v$.
If the path from $u$ to $v$ containing $k$ edges, color $c$ is called dominating if it appears at least $\lceil \frac{k}{2} \rceil + 1$ times.
### Input
- The first line contains two integers $n, q$.
- The second line contains $n$ integers $A_i$.
- The next $n-1$ lines, each line contains two integers $u, v$, there is an edge between $u$ and $v$.
- The next $q$ lines, each line contains two integers $u,v$, a query.
### Output
- Print the answer for each query, or print `-1` if no answer exists.
### Constraints
- $1 \le n \le 10^5$.
- $1 \le A_i \le 10^9$.
- $1 \le u, v \le n$.
### Example
Input:
```
4 3
1 2 2 2
1 2
2 3
2 4
1 2
2 3
1 4
```
Output:
```
-1
2
2
```
