Problem C
Point in Polygon

You are given a simple polygon and a query point, and you wish to determine whether the query point is inside or outside the polygon.

In the 1st, 3rd, and 4th example below, the query point $q$ is “inside”; in the others it is “outside”.

Input

The input consists of:

• One line with the coordinates $q_x$ and $q_y$ of the query point.

• One line with the number $n$ of corners of the polygon, where $3\leq n\leq 100\, 000$.

• $n$ lines, the $i$th of which contains the coordinates $x_i$ and $y_i$ of the $i$th corner for $1\leq i \leq n$.

Each coordinate is given as a floating point number. The $x$-coordinates satisfy $0\leq q_x, x_1,\ldots ,x_n\leq 180$. The $y$-coordinates satisfy $0\leq q_y, y_1,\ldots ,y_n\leq 90$.

It is guaranteed that the points are all different and that $(x_1,y_1),\ldots , (x_n,y_n), (x_1,y_1)$ (in this order) is a closed polygonal chain that forms the boundary of a simple polygon. Moreover, the query point $q$ has distance at least $10^{-6}$ from the boundary. Consequently, the query point is guaranteed to lie strictly inside or strictly outside the polygon.

Output

Write “inside” if $q$ is inside the polygon. Else write “outside”.

1.0 1.0
4
0.0 0.0
2.0 0.0
2.0 2.0
0.0 2.0

inside

0.0 0.0
4
1.0 0.0
2.0 1.0
1.0 2.0
0.0 1.0

outside

1.0 1.0
6
0.0 0.0
2.0 0.0
2.0 1.0
3.0 1.0
3.0 2.0
0.0 2.0

inside

1.0 1.0
4
1.0 0.0
2.0 1.0
1.0 2.0
0.0 1.0

inside

1.0 1.0
8
1.0 0.0
2.0 1.0
1.0 2.0
0.0 1.0
0.25 0.75
1.0 1.5
1.5 1.0
0.75 0.25

outside