P9672 [ICPC 2022 Jinan R] Grid Points

You are given a simple polygon in the first quadrant of the Cartesian plane. This means that the coordinate $(x,y)$ of any point in the polygon satisfies $x> 0$ and $y> 0$. Consider all grid points in the polygon. Order them in increasing order of $\textbf{slopes}$. Output the $k$-th grid point in this order. A grid point is a point $(x, y)$ such that $x$ and $y$ are integers. A point on the boundary of a polygon is considered to be in the polygon. The slope of point $(x, y)$ is $y/x$. If two points have the same slope, they are ordered lexicographically first by $x$, then by $y$. In other words, a point $(x_1, y_1)$ is lexicographically less than $(x_2, y_2)$ if $(x_1

The first line contains one integer $T~(1\le T \le 500)$, the number of test cases. For each test case, the first line contains two integers $n, k~(n\ge 3, 1\le k)$. Each of the next $n$ lines describes a vertex of the polygon. Each vertex is denoted by two integers $x, y~(1\le x, y\le 10^9)$, representing its coordinate $(x, y)$. The vertices are given in counterclockwise order. It is guaranteed that the polygon is simple, i.e.~ the vertices are distinct, two edges may overlap only when they are consecutive on the boundary, and the overlap contains exactly $1$ point. It is guaranteed that $k$ is no more than the number of grid points in the polygon. It is guaranteed that the sum of $n$ over all test cases is no more than $2000$.

For each test case, output two integers $x, y$ in one line representing the coordinate $(x, y)$ of the $k$-th grid point.