Yet Another Angular Sort Problem

There are $N$ distinct points $A_1,A_2,\cdots,A_N$ in the plane, and no three of them are collinear. A line $\ell$ passing through none of the points is also given. A point $P$ moves along $\ell$ from infinity on one side to infinity on the other side.

At every moment, sort $A_1,A_2,\cdots,A_N$ around $P$. Starting from a ray parallel to $\ell$ in one fixed direction, list the point indices encountered clockwise. Call the resulting permutation $Q$.

The permutation may change several times. If $Q$ and $Q'$ are the permutations immediately before and after one change, define its amount as

Find the sum of the amounts of all changes modulo $998\,244\,353$.

The input is given in the following format:

$T$ $\mathrm{case}_1$ $\mathrm{case}_2$ $\vdots$ $\mathrm{case}_T$

Each case is given in the following format:

$N$ $a\ b\ c$ $x_1\ y_1$ $x_2\ y_2$ $\vdots$ $x_N\ y_N$

The equation of $\ell$ is $ax+by+c=0$.

For each test case, print the sum of all change amounts modulo $998\,244\,353$.

• $1 \leq T \leq 100$
• $1 \leq N \leq 300\,000$
• $-10^9 \leq a,b,c,x_i,y_i \leq 10^9$
• $(a,b) \neq (0,0)$
• All points are distinct, and no three points are collinear.
• The line $\ell$ passes through none of the given points.
• The sum of $N$ over all test cases does not exceed $300\,000$.