Problem:
Score : 700 points

Problem Statement

Takahashi has a tower which is divided into N layers.
Initially, all the layers are uncolored. Takahashi is going to paint some of the layers in red, green or blue to make a beautiful tower.
He defines the beauty of the tower as follows:

The beauty of the tower is the sum of the scores of the N layers, where the score of a layer is A if the layer is painted red, A+B if the layer is painted green, B if the layer is painted blue, and 0 if the layer is uncolored.

Here, A and B are positive integer constants given beforehand. Also note that a layer may not be painted in two or more colors.

Takahashi is planning to paint the tower so that the beauty of the tower becomes exactly K.
How many such ways are there to paint the tower? Find the count modulo 998244353.
Two ways to paint the tower are considered different when there exists a layer that is painted in different colors, or a layer that is painted in some color in one of the ways and not in the other.

Constraints

1 ≤ N ≤ 3×10^5

1 ≤ A,B ≤ 3×10^5

0 ≤ K ≤ 18×10^{10}

All values in the input are integers.

Input

Input is given from Standard Input in the following format:

N A B K

Output

Print the number of the ways to paint tiles, modulo 998244353.

Sample Input 1

4 1 2 5

Sample Output 1

40

In this case, a red layer worth 1 points, a green layer worth 3 points and the blue layer worth 2 points. The beauty of the tower is 5 when we have one of the following sets of painted layers:

1 green, 1 blue

1 red, 2 blues

2 reds, 1 green

3 reds, 1 blue

The total number of the ways to produce them is 40.

Sample Input 2

2 5 6 0

Sample Output 2

1

The beauty of the tower is 0 only when all the layers are uncolored. Thus, the answer is 1.

Sample Input 3

90081 33447 90629 6391049189

Sample Output 3

577742975

Solution in Fortran (gfortran v4.8.4) (judge verdict: Accepted)

Program main
  implicit none
  integer(8) :: n, a,b, K
  integer(8) :: a_1, b_1
  integer(8) :: i, j
  integer(8) :: mbase
  integer(8),allocatable :: f(:),finv(:),inv(:)
  integer :: out
  mbase = 998244353
  read(*,*) n, a, b, K
  allocate( f(0:N), finv(0:N), inv(0:N) )
  f(0) = 1
  f(1) = 1
  finv(0) = 1
  finv(1) = 1
  inv(0) = 0
  inv(1) = 1
  do i = 2,N
     f(i) = modulo( f(i-1)* i,mbase  )
     inv(i) = modulo ( -inv(modulo(mbase,i)) *(mbase/ i ), mbase )
     finv(i) = modulo( finv(i-1)*inv(i), mbase ) 
  end do
  out = 0
  do i = 0, N
     if ( mod( ( K - (A*i) ),B) == 0 ) then
        j = ( K - (A*i) )/B
        if ( 0 <= j.and.j <= N  ) then
           out = modulo(modulo( nCr2(n,i,finv,f) * nCr2(n,j,finv,f) , mbase) + out, mbase)
        end if
     end if
  end do
  write(*,'(i0)') out
  stop
contains
  integer(8) function mpow(a,n,mbase )
    ! a**n mod mbase
    integer(8) :: a
    integer(8) ::  n, mbase
    integer(8) :: ntmp, atmp
    mpow = 1
    atmp = a
    ntmp = n
    do while(ntmp > 0)
       !if ( mod(ntmp,2) == 0   ) mpow = modulo( mpow * atmp,mbase)
       if ( iand(ntmp,1) >0 ) mpow = modulo( mpow * atmp,mbase)
       atmp = modulo( atmp * atmp, mbase )
       ntmp = ishft(ntmp,-1)
    end do
  end function mpow

  
  integer(8) function nCr(n,r )
    integer(8) :: n, r
    integer(8) :: n_r
    integer(8) :: i
    integer(8) :: n1, n2
    integer(8) :: inv, inv2
    if ( r == 0 ) then
       nCr = 1
       return
    end if
    n_r = n-r
    n1 = 1
    do i = n,n_r+1,-1
       n1 =  mod(n1*i,mbase)
    end do
    n2 = 1
    do i = 1,r
       n2 = mod(n2*i,mbase)
    end do
    n2 = mpow(n2,mbase-2,mbase)
    nCr =  mod( n1*n2,mbase)
  end function nCr
  
  integer(8) function nCr2(n,r,finv,f)
    integer(8) :: finv(0:N), f(0:N)
    integer(8) :: n,r, r2
    r2 = min(r,n-r)
    !nCr = modulo( f(n)*modulo( (finv(r2)*finv(n-r2)), mbase ), mbase )
    nCr2 = mod( f(n)*(mod(finv(r2)*finv(n-r2),mbase) ), mbase )
  end function nCr2
  
end program main