P15768 [JAG 2025 Summer Camp #2] Triangles

You are given $N$ distinct points on a 2D plane. The $i$-th point is located at $(x_i, y_i)$. For each integer $k \geq 1$, let $f(k)$ be the maximum number of non-degenerate triangles you can place under the following conditions: - You add $k$ new points on the plane such that all $N + k$ points are distinct. - Each triangle has its vertices among the $N + k$ points. - No two triangles have an intersection with a positive area. Compute $\left(\sum \limits_{k=1}^{K} f(k)\right) \mod 998244353$.

The input consists of a single test case in the following format. $$ \begin{aligned} & N \ K \\ & x_1 \ y_1 \\ & \vdots \\ & x_N \ y_N \end{aligned} $$ The first line contains two integers $N$ and $K$ ($1 \leq N \leq 200\,000$, $1 \leq K \leq 10^9$), representing the number of points and the maximum value of $k$. Each of the next $N$ lines contains two integers $x_i$ and $y_i$ ($0 \leq x_i, y_i \leq 10^9$), representing the coordinates of the $i$-th point. It is guaranteed that all $N$ points are distinct.

Print the answer.