CSES - Point in Polygon
  • Time limit: 1.00 s
  • Memory limit: 512 MB
You are given a polygon of $n$ vertices and a list of $m$ points. Your task is to determine for each point if it is inside, outside or on the boundary of the polygon.

The polygon consists of $n$ vertices $(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n)$. The vertices $(x_i,y_i)$ and $(x_{i+1},y_{i+1})$ are adjacent for $i=1,2,\dots,n-1$, and the vertices $(x_1,y_1)$ and $(x_n,y_n)$ are also adjacent.

Input

The first input line has two integers $n$ and $m$: the number of vertices in the polygon and the number of points.

After this, there are $n$ lines that describe the polygon. The $i$th such line has two integers $x_i$ and $y_i$.

You may assume that the polygon is simple, i.e., it does not intersect itself.

Finally, there are $m$ lines that describe the points. Each line has two integers $x$ and $y$.

Output

For each point, print "INSIDE", "OUTSIDE" or "BOUNDARY".

Constraints
  • $3 \le n,m \le 1000$
  • $1 \le m \le 1000$
  • $-10^9 \le x_i, y_i \le 10^9$
  • $-10^9 \le x, y \le 10^9$
Example

Input:
4 3
1 1
4 2
3 5
1 4
2 3
3 1
1 3


Output:
INSIDE
OUTSIDE
BOUNDARY