201334: [AtCoder]ARC133 E - Cyclic Medians

Problem Statement

Given are integers $N$, $M$, $V$, and $A$. Consider the following procedure.

• Choose a sequence of $N$ integers between $1$ and $V$ (inclusive): $x=(x_1,x_2,\cdots,x_N)$.
• Choose a sequence of $M$ integers between $1$ and $V$ (inclusive): $y=(y_1,y_2,\cdots,y_M)$.
• Let $a$ be a variable and initialize it with $a=A$.
• For each $i=0,1,\cdots,N \times M-1$, do the following.
  • Replace the value of $a$ with the median of $a,x_{(i \bmod N)+1},y_{(i \bmod M)+1}$.
• Print the final value of $a$.

Consider doing this procedure with every possible pair of sequences $x,y$. Find the sum of the values that will be printed, modulo $998244353$.

Constraints

• $1 \leq N,M \leq 200000$
• $1 \leq A \leq V \leq 200000$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $V$ $A$

Output

Print the answer.

Sample Input 1

2 1 2 1

Sample Output 1

11

For example, when $x=(1,2),y=(2)$, the procedure goes as follows.

• Initialize $a$ with $a=1$.
• For $i=1$: replace the value of $a$ with the median of $a(=1),x_1(=1),y_1(=2)$, which is $1$.
• For $i=2$: replace the value of $a$ with the median of $a(=1),x_2(=2),y_1(=2)$, which is $2$.
• Print $a(=2)$.

There are three cases where $2$ will be printed: $(x=(1,2),y=(2))$, $(x=(2,1),y=(2))$, $(x=(2,2),y=(2))$. In the other five cases, $1$ will be printed. Therefore, the answer is $2 \times 3 + 1\times 5=11$.

Sample Input 2

2 2 5 4

Sample Output 2

2019

Sample Input 3

2100 2300 2201 2022

Sample Output 3

407723438