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 Haskell (GHC 7.10.3) (judge verdict: Accepted)

import qualified Data.Text as T
import qualified Data.Text.IO as T
import qualified Data.Text.Read as T

main :: IO ()
main = do
  (n : _) <- map unsafeSignedDecimal . T.words <$> T.getLine :: IO [Int]
  as <- map unsafeSignedDecimal . T.words <$> T.getLine :: IO [Int]
  print . fst . foldl step (1, (0, 0, 0)) $ as

step :: (Int, (Int, Int, Int)) -> Int -> (Int, (Int, Int, Int))
step (k, cs@(c0, c1, c2)) a = ((k * (length . filter (== True) $ [b0, b1, b2])) `mod` modulus, cs')
  where
    b0 = c0 == a
    b1 = c1 == a
    b2 = c2 == a
    cs' | b0 = (succ c0, c1, c2)
        | b1 = (c0, succ c1, c2)
        | b2 = (c0, c1, succ c2)
        | otherwise = cs

modulus = 1000000007 :: Int

unsafeSignedDecimal :: T.Text -> Int
unsafeSignedDecimal s = case T.signed T.decimal s of
  Right (n, _) -> n