Problem:
Score : 800 points

Problem Statement

Snuke has a string x of length N.
Initially, every character in x is 0.

Snuke can do the following two operations any number of times in any order:

Choose A consecutive characters in x and replace each of them with 0.

Choose B consecutive characters in x and replace each of them with 1.

Find the number of different strings that x can be after Snuke finishes doing operations.
This count can be enormous, so compute it modulo (10^9+7).

Constraints

1 \leq N \leq 5000

1 \leq A,B \leq N

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N A B

Output

Print the number of different strings that x can be after Snuke finishes doing operations, modulo (10^9+7).

Sample Input 1

4 2 3

Sample Output 1

11

For example, x can be 0011 or 1111 in the end, but cannot be 0110.

Sample Input 2

10 7 2

Sample Output 2

533

Sample Input 3

1000 100 10

Sample Output 3

828178524

Solution in Fortran (GNU Fortran 9.2.1) (judge verdict: Accepted)

module mod_modint
  implicit none
  integer(8) :: modulus = 10_8**9+7
  type modint
    private
    integer(8) :: num
  contains
    procedure :: get => getnum
  end type

  interface assignment(=)
    module procedure :: setm, seti64, seti32
  end interface assignment(=)

  interface operator(+)
    module procedure :: posm, addmm, addmi64, addmi32, addi64m, addi32m
  end interface operator(+)

  interface operator(-)
    module procedure :: negm, submm, submi64, submi32, subi64m, subi32m
  end interface operator(-)

  interface operator(*)
    module procedure :: mulmm, mulmi64, mulmi32, muli64m, muli32m
  end interface operator(*)

  interface operator(/)
    module procedure :: divmm, divmi64, divmi32, divi64m, divi32m
  end interface operator(/)

  interface operator(**)
    module procedure :: powmi64, powmi32
  end interface operator(**)

  interface inv
    module procedure :: invm
  end interface inv

contains

  integer(8) function getnum(this)
    class(modint), intent(in) :: this
    getnum = this%num
  end

  pure elemental type(modint) function newm()
    newm%num = 0_8
  end

  pure elemental subroutine setm(x,y)
    type(modint), intent(inout) :: x
    type(modint), intent(in) :: y
    x%num = y%num
  end

  pure elemental function posm(x) result(n)
    type(modint), intent(in) :: x
    type(modint) :: n
    n%num = x%num
  end

  pure elemental function negm(x) result(n)
    type(modint), intent(in) :: x
    type(modint) :: n
    n%num = modulus-x%num
  end

  pure elemental function addmm(x,y) result(n)
    type(modint), intent(in) :: x, y
    type(modint) :: n
    n%num = x%num+y%num
    if (n%num >= modulus) n%num = n%num-modulus
  end

  pure elemental function submm(x,y) result(n)
    type(modint), intent(in) :: x, y
    type(modint) :: n
    n%num = x%num-y%num
    if (n%num < 0_8) n%num = n%num+modulus
  end

  pure elemental function mulmm(x,y) result(n)
    type(modint), intent(in) :: x, y
    type(modint) :: n
    n%num = mod(x%num*y%num,modulus)
  end

  impure elemental function invm(x) result(n)
    type(modint), intent(in) :: x
    type(modint) :: n
    integer(8) :: a, b, c, q, r, v
    a = x%num
    if (a == 0_8) then
      write(*,'(a)') "Error: Division by zero (x == 0). (modint, invm)"
      stop
    end if
    b = modulus
    c = 0_8
    v = 1_8
    do while (b /= 0_8)
      q = a/b
      r = mod(a,b)
      a = b
      b = r
      r = c
      c = v-c*q
      v = r
    end do
    n%num = mod(v,modulus)
    if (n%num < 0_8) n%num = n%num+modulus
  end

  impure elemental function divmm(x,y) result(n)
    type(modint), intent(in) :: x, y
    type(modint) :: n
    n = mulmm(x,invm(y))
  end

  impure elemental type(modint) function newi(i)
    class(*), intent(in) :: i
    select type(i)
    type is (integer(8))
      newi%num = i
    type is (integer)
      newi%num = int(i,8)
    type is (integer(2))
      newi%num = int(i,8)
    type is (integer(1))
      newi%num = int(i,8)
    class default
      write(*,'(a)') "Error: Invalid value (i is not integer). (modint, newi)"
      stop
    end select
    newi%num = mod(newi%num,modulus)
    if (newi%num < 0_8) newi%num = newi%num+modulus
  end

  impure elemental subroutine seti64(x,i)
    type(modint), intent(inout) :: x
    integer(8), intent(in) :: i
    call setm(x,newi(i))
  end

  impure elemental subroutine seti32(x,i)
    type(modint), intent(inout) :: x
    integer, intent(in) :: i
    call setm(x,newi(i))
  end

  impure elemental function addmi64(x,i) result(n)
    type(modint), intent(in) :: x
    integer(8), intent(in) :: i
    type(modint) :: n
    n = addmm(x,newi(i))
  end

  impure elemental function addmi32(x,i) result(n)
    type(modint), intent(in) :: x
    integer, intent(in) :: i
    type(modint) :: n
    n = addmm(x,newi(i))
  end

  impure elemental function addi64m(i,y) result(n)
    integer(8), intent(in) :: i
    type(modint), intent(in) :: y
    type(modint) :: n
    n = addmm(newi(i),y)
  end

  impure elemental function addi32m(i,y) result(n)
    integer, intent(in) :: i
    type(modint), intent(in) :: y
    type(modint) :: n
    n = addmm(newi(i),y)
  end

  impure elemental function submi64(x,i) result(n)
    type(modint), intent(in) :: x
    integer(8), intent(in) :: i
    type(modint) :: n
    n = submm(x,newi(i))
  end

  impure elemental function submi32(x,i) result(n)
    type(modint), intent(in) :: x
    integer, intent(in) :: i
    type(modint) :: n
    n = submm(x,newi(i))
  end

  impure elemental function subi64m(i,y) result(n)
    integer(8), intent(in) :: i
    type(modint), intent(in) :: y
    type(modint) :: n
    n = submm(newi(i),y)
  end

  impure elemental function subi32m(i,y) result(n)
    integer, intent(in) :: i
    type(modint), intent(in) :: y
    type(modint) :: n
    n = submm(newi(i),y)
  end

  impure elemental function mulmi64(x,i) result(n)
    type(modint), intent(in) :: x
    integer(8), intent(in) :: i
    type(modint) :: n
    n = mulmm(x,newi(i))
  end

  impure elemental function mulmi32(x,i) result(n)
    type(modint), intent(in) :: x
    integer, intent(in) :: i
    type(modint) :: n
    n = mulmm(x,newi(i))
  end

  impure elemental function muli64m(i,y) result(n)
    integer(8), intent(in) :: i
    type(modint), intent(in) :: y
    type(modint) :: n
    n = mulmm(newi(i),y)
  end

  impure elemental function muli32m(i,y) result(n)
    integer, intent(in) :: i
    type(modint), intent(in) :: y
    type(modint) :: n
    n = mulmm(newi(i),y)
  end

  impure elemental function divmi64(x,i) result(n)
    type(modint), intent(in) :: x
    integer(8), intent(in) :: i
    type(modint) :: n
    n = divmm(x,newi(i))
  end

  impure elemental function divmi32(x,i) result(n)
    type(modint), intent(in) :: x
    integer, intent(in) :: i
    type(modint) :: n
    n = divmm(x,newi(i))
  end

  impure elemental function divi64m(i,y) result(n)
    integer(8), intent(in) :: i
    type(modint), intent(in) :: y
    type(modint) :: n
    n = divmm(newi(i),y)
  end

  impure elemental function divi32m(i,y) result(n)
    integer, intent(in) :: i
    type(modint), intent(in) :: y
    type(modint) :: n
    n = divmm(newi(i),y)
  end

  impure elemental function powmi64(x,i) result(n)
    type(modint), intent(in) :: x
    integer(8), intent(in) :: i
    type(modint) :: n, p
    integer(8) :: m
    n = newi(1_8)
    p = x
    m = i
    if (i < 0_8) then
      p = invm(x)
      m = abs(i)
    end if
    do while (m > 0_8)
      if (btest(m,0)) n = mulmm(p,n)
      p = mulmm(p,p)
      m = rshift(m,1)
    end do
  end

  impure elemental function powmi32(x,i) result(n)
    type(modint), intent(in) :: x
    integer, intent(in) :: i
    type(modint) :: n, p
    integer :: m
    n = newi(1_8)
    p = x
    m = i
    if (i < 0) then
      p = invm(x)
      m = abs(i)
    end if
    do while (m > 0)
      if (btest(m,0)) n = mulmm(p,n)
      p = mulmm(p,p)
      m = rshift(m,1)
    end do
  end

end module mod_modint
program c
   use mod_modint
   implicit none
   integer::N
   integer::a,b
   type(modint),allocatable,dimension(:,:)::f,DP
   type(modint),allocatable,dimension(:)::g
   type(modint)::w,ANS,tmp
   integer::i,j,k
   read*,N,a,b
   if(a>b)then
      a=xor(a,b)
      b=xor(a,b)
      a=xor(a,b)
   endif
   allocate(f(0:N,2),DP(0:N,2))
   allocate(g(0:N))

   f=0
   DP=0
   ANS=0

   if(b>=2)then
      f(0,1)=1;f(0,2)=1
      do i=1,N
         do j=1,2
            do k=1,i
               if(j==1.and.k<a)cycle
               f(i,j)=f(i,j)+f(i-k,mod(j,2)+1)
            end do
         end do
      end do
      g(0)=1
      do i=1,N
         g(i)=f(i,1)+f(i,2)
      end do
      do i=1,N-1
         if(i<a)dp(i,1)=dp(i,1)+1
         do j=1,min(i,a)-1
            dp(i,1)=dp(i,1)+dp(i-j,2)
         end do
         if(i<b)dp(i,2)=dp(i,2)+g(i-1)
         do j=1,min(i,b)-1
            w=1
            if(j/=1)w=g(j-2)
            dp(i,2)=dp(i,2)+dp(i-j,1)*w
         end do
         if(n-i<b)ans=ans+dp(i,1)*g(n-i-1)
         if(n-i<a)ans=ans+dp(i,2)
      end do
   endif
   tmp=2
   tmp=tmp**N
   print*,getnum(tmp-ans)

end program c