Problem:
Score : 600 points

Problem Statement

We have a grid with A horizontal rows and B vertical columns, with the squares painted white. On this grid, we will repeatedly apply the following operation:

Assume that the grid currently has a horizontal rows and b vertical columns. Choose "vertical" or "horizontal".

If we choose "vertical", insert one row at the top of the grid, resulting in an (a+1) \times b grid.

If we choose "horizontal", insert one column at the right end of the grid, resulting in an a \times (b+1) grid.

Then, paint one of the added squares black, and the other squares white.

Assume the grid eventually has C horizontal rows and D vertical columns. Find the number of ways in which the squares can be painted in the end, modulo 998244353.

Constraints

1 \leq A \leq C \leq 3000

1 \leq B \leq D \leq 3000

A, B, C, and D are integers.

Input

Input is given from Standard Input in the following format:

A B C D

Output

Print the number of ways in which the squares can be painted in the end, modulo 998244353.

Sample Input 1

1 1 2 2

Sample Output 1

3

Any two of the three squares other than the bottom-left square can be painted black.

Sample Input 2

2 1 3 4

Sample Output 2

65

Sample Input 3

31 41 59 265

Sample Output 3

387222020

Solution in Kotlin (1.3.71) (judge verdict: Time Limit Exceeded)

import java.io.*
import java.lang.*


class LongMod(var num: Long, val MOD: Long = 1_000_000_007L) {
    init {
        if (MOD <= 0) throw Error("MOD <= 0")
        this.num = (this.num%this.MOD + this.MOD)%this.MOD
    }
    constructor(numStr: String, Base: Int=10, MOD: Long=1_000_000_007L) :
            this(0L, MOD) {
        if (numStr.isEmpty()) return
        var rv = numStr.reversed()
        var d = 1L
        rv.forEach{ c ->
            this.num += (c - '0') * d
            this.num %= MOD
            d *= Base
            d %= MOD
        }
    }
    override operator fun equals(other: Any?): Boolean {
        if (other !is LongMod) return false
        return this.num % this.MOD == other.num % this.MOD
    }
    operator fun unaryMinus(): LongMod {
        return LongMod(-this.num)
    }
    private operator fun plusAssign(other: LongMod): Unit {
        if ((this.num + other.num) >= this.MOD) {
            this.num -= this.MOD
        }
        this.num += other.num
    }
    private operator fun minusAssign(other: LongMod): Unit {
        if ((this.num + this.MOD - other.num) >= this.MOD) {
            this.num -= this.MOD
        }
        this.num += this.MOD - other.num
    }
    private operator fun timesAssign(other: LongMod): Unit {
        this.num *= other.num
        this.num %= this.MOD
    }
    operator fun plus(other: LongMod): LongMod {
        val res = LongMod(this.num, this.MOD)
        res += other
        return res
    }
    operator fun minus(other: LongMod): LongMod {
        val res = LongMod(this.num, this.MOD)
        res -= other
        return res
    }
    operator fun times(other: LongMod): LongMod {
        val res = LongMod(this.num, this.MOD)
        res *= other
        return res
    }
    fun pow(n: Long): LongMod {
        if (n == 0L) return LongMod(1L, this.MOD)
        var x = this.pow(n / 2)
        x.timesAssign(x)
        if (n % 2 == 1L) x.timesAssign(this)
        return x
    }
    fun inv(): LongMod {
        return this.pow(this.MOD - 2)
    }
    operator fun divAssign(other: LongMod): Unit {
        this *= other.inv()
    }
    operator fun div(other: LongMod): LongMod {
        val res = LongMod(this.num, this.MOD)
        res /= other
        return res
    }
}


fun solve() {
    val (A,B,C,D) = rd.readIntArray()

    val Mod = 998244353L
    val dp = Array(C+1){Array(D+1){LongMod(0L, Mod)}}
    dp[A][B] = LongMod(1L, Mod)

    for (i in A..C) {
        val mi = LongMod(i.toLong(), Mod)
        val mi1 = LongMod((i-1).toLong(), Mod)
        for (j in B..D) {
            dp[i][j] += dp[i-1][j] * LongMod(j.toLong(), Mod)
            dp[i][j] += dp[i][j-1] * mi
            dp[i][j] -= dp[i-1][j-1] * LongMod((j-1).toLong(), Mod) * mi1
        }
    }

    val ans = dp[C][D].num
    println(ans)
}


//val rd = debug.Reader("agc", "046", "b", "sample-1")
val rd = Reader()


fun main(args: Array<String>) {
    try {
        var exception: Throwable? = null
        val sth = Thread(null, SolveThread(), "solveThread", 128 * 1024 * 1024)
        sth.setUncaughtExceptionHandler { _, e -> exception = e }
        sth.start(); sth.join()
        if (exception != null) throw exception!!
    } catch (e: Exception) { throw e }
}
const val MOD = 1_000_000_007L; const val INF = Int.MAX_VALUE/2; const val LINF = Long.MAX_VALUE/2
data class IntPair(val first: Int, val second: Int); data class LongPair(val first: Long, val second: Long)
val errPrintln: (Any) -> Unit = { msg -> System.err.println(msg) }
fun rAssert(exp: Boolean, lazyMsg: () -> String = {""}) {
    if (!exp) { val msg = lazyMsg(); errPrintln("EAssert Error: $msg"); throw Error(msg) }
}
fun tAssert(exp: Boolean, lazyMsg: () -> String = {""}) {
    if (!exp) { val msg = lazyMsg(); errPrintln("TAssert Error: $msg"); Thread.sleep(3000) }
}
class SolveThread : Runnable {
    override fun run() { solve() }
}
class Reader {
    private val br = BufferedReader(InputStreamReader(System.`in`))
    val readString: () -> String = { br.readLine() }
    val readInt: () -> Int = { br.readLine().toInt() }
    val readLong: () -> Long = { br.readLine().toLong() }
    val readIntArray: () -> IntArray = { br.readLine().split(' ').map { it.toInt() }.toIntArray() }
    val readLongArray: () -> LongArray = { br.readLine().split(' ').map { it.toLong() }.toLongArray() }
    val readListInt: () -> List<Int> = { br.readLine().split(' ').map { it.toInt() } }
    val readListLong: () -> List<Long> = { br.readLine().split(' ').map { it.toLong() } }
}