Problem:
Score: 500 points

Problem Statement

N people are standing in a queue, numbered 1, 2, 3, ..., N from front to back. Each person wears a hat, which is red, blue, or green.

The person numbered i says:

"In front of me, exactly A_i people are wearing hats with the same color as mine."

Assuming that all these statements are correct, find the number of possible combinations of colors of the N people's hats.

Since the count can be enormous, compute it modulo 1000000007.

Constraints

1 \leq N \leq 100000

0 \leq A_i \leq N-1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 A_3 ... A_N

Output

Print the number of possible combinations of colors of the N people's hats, modulo 1000000007.

Sample Input 1

6
0 1 2 3 4 5

Sample Output 1

3

We have three possible combinations, as follows:

Red, Red, Red, Red, Red, Red

Blue, Blue, Blue, Blue, Blue, Blue

Green, Green, Green, Green, Green, Green

Sample Input 2

3
0 0 0

Sample Output 2

6

Sample Input 3

54
0 0 1 0 1 2 1 2 3 2 3 3 4 4 5 4 6 5 7 8 5 6 6 7 7 8 8 9 9 10 10 11 9 12 10 13 14 11 11 12 12 13 13 14 14 15 15 15 16 16 16 17 17 17

Sample Output 3

115295190

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

program colorful_hats_2
  implicit none
  integer(8), parameter :: md = 1000000007_8
  integer :: n, a(100000) = 0, i, c(3) = 0, j
  integer(8) :: ans = 1_8, m
  read(*,*) n
  read(*,*) a(1:n)
  do i = 1, n
    m = 0_8
    do j = 1, 3
      if (a(i) == c(j)) m = m+1_8
    end do
    ans = mod(ans*m,md)
    do j = 1, 3
      if (a(i) == c(j)) then
        c(j) = c(j)+1
        exit
      end if
    end do
  end do
  write(*,'(i0)') ans
end program colorful_hats_2