Split Game

Consider the following game about splitting a simple polygon with $N$ vertices on a plane. The purpose of this game is using a straight line which passes through the origin to split the given simple polygon into as many non-zero area regions as possible. Please finish the game with the best result possible.

The input consists of $N+1$ lines. The first line contains an integer $N$. The $i$-th of the following $N$ lines consists of two integers $x\_i$ and $y\_i$ indicating the vertices of the given polygon in counter-clockwise order.

Output one integer: the maximum number of non-zero area regions into which the given polygon can be split by a single line passing through the origin.

•  $1 \le N \le 10^5$ 
•  $1 \le x\_i, y\_i \le 10^9$ 
• if $i \ne j$, then $(x\_i, y\_i) \ne (x\_j, y\_j)$ 
• the vertices are given in counter-clockwise order