Cyclic Medians

AtCoder

IDarc133_e

Time2000ms

Memory256MB

Difficulty—

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: $N$ $M$ $V$ $A$ [samples]

Samples

Input #1

2 1 2 1

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 $2 \times 3 + 1\times 5=11$.

Input #2

2 2 5 4

Output #2

Input #3

2100 2300 2201 2022

Output #3

407723438

API Response (JSON)

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    "name": "Cyclic Medians",
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      "content": "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 ",
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      "statement_type": "Markdown",
      "content": "Given are integers $N$, $M$, $V$, and $A$. Consider the following procedure.\n\n*   Choose a sequence of $N$ integers between $1$ and $V$ (inclusive): $x=(x_1,x_2,\\cdots,x_N)$.\n*   Choose a sequence of ...",
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      "language": "English"
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