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 Ruby (2.3.3) (judge verdict: Accepted)

n = gets.to_i
a = gets.split.map(&:to_i)
if a[0] != 0
  puts 0
  exit
end

mod = 1000000007
cnt = Array.new(n + 1, 0)
cnt[0] = 1
ans = 3

1.upto(n - 1) do |i|
  if a[i] != 0 && cnt[a[i] - 1] <= cnt[a[i]]
    ans *= 0
  end
  if cnt[a[i]] == 0 # 1回めの出現
    ans *= cnt[a[i] - 1]
    ans %= mod
    cnt[a[i]] += 1
  elsif cnt[a[i]] == 1
    if a[i] == 0
      ans *= 2
    else
      ans *= (cnt[a[i] - 1] - cnt[a[i]])
    end
    ans %= mod
    cnt[a[i]] += 1
  elsif cnt[a[i]] == 2
    cnt[a[i]] += 1
  else
    ans *= 0
  end
end
p ans