P5170 [Template] Euclidean-like Algorithm

Given $n,\,a,\,b,\,c$, compute respectively $\sum\limits_{i=0}^{n}\left\lfloor \frac{ai+b}{c} \right\rfloor$, $\sum\limits_{i=0}^{n}{\left\lfloor \frac{ai+b}{c} \right\rfloor}^2$, and $\sum\limits_{i=0}^{n} i \left\lfloor \frac{ai+b}{c} \right\rfloor$. Output each answer modulo $998244353$. There are multiple test cases.

The first line contains the number of test cases $t$. The next $t$ lines each contain four integers, which are $n,\ a,\ b,\ c$ for that test case.

For each test case, output one line with three integers, which are the three answers modulo $998244353$.

This problem uses $\mathrm{Special}\ \mathrm{Judge}$. If you answer all cases for the first query correctly, you can get $40\%$ of the total score. If you answer all cases for the second and third queries correctly, you can get an additional $30\%$ for each. |Test Point ID|Special Property| |:-:|:-:| |$1$|$n,\,a,\,b,\,c\leqslant10$| |$2 \sim 3$|$n,\,a,\,b,\,c\leqslant1000$| |$4$|$n,\,a,\,b,\,c\leqslant10^6$| |$5$|$t=1$| |$6 \sim 10$|None| For all test points, $1 \leqslant t \leqslant 10^5,\ 0 \leqslant n,\,a,\,b,\,c \leqslant 10^9,\ c \neq 0$. Translated by ChatGPT 5