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 Lua (LuaJIT 2.1.0) (judge verdict: Accepted)

local mod = 998244353
local function badd(x, y) return (x + y) % mod end
local function bmul(x, y) return (x * y) % mod end
local a, b = io.read("*n", "*n")
local c, d = io.read("*n", "*n")
local dp = {}
for i = 1, c do
  dp[i] = {}
  for j = 1, d do
    dp[i][j] = 0
  end
end
dp[a][b] = 1
for ia = a, c do
  for ib = b, d do
    if ia == a then
      if ib == b then
      else
        dp[a][ib] = bmul(dp[a][ib - 1], a)
      end
    elseif ib == b then
      dp[ia][b] = bmul(dp[ia - 1][b], b)
    else
      local v1 = bmul(dp[ia][ib - 1], ia)
      local v2 = bmul(dp[ia - 1][ib], ib)
      local v3 = bmul(dp[ia - 1][ib - 1], bmul(ia - 1, ib - 1))
      v3 = mod - v3
      dp[ia][ib] = badd(v1, badd(v2, v3))
    end
  end
end
print(dp[c][d])