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 OCaml (4.02.3) (judge verdict: Accepted)

let m = 1000000007
let ( +^ ) x y = (x + y) mod m
let ( *^ ) x y = (x * y) mod m

module IntTripleMap = Map.Make (struct
  type t = int * int * int
  let compare = compare
end)

let () = Scanf.scanf "%d\n" @@ fun n ->
  let as_ = Array.init n @@ fun _ -> Scanf.scanf "%d " @@ fun a -> a in
  let m =
    Array.fold_left (fun m a ->
      IntTripleMap.fold (fun (r, g, b) n m ->
        let m =
          if r <> a then m else 
          IntTripleMap.add (r + 1, g, b) 
          (n +^
            try IntTripleMap.find (r + 1, g, b) m
            with Not_found -> 0) m in
        let m =
          if g <> a then m else 
          IntTripleMap.add (r, g + 1, b) 
            (n +^
              try IntTripleMap.find (r, g + 1, b) m
              with Not_found -> 0) m in
        let m =
          if b <> a then m else 
          IntTripleMap.add (r, g, b + 1) 
            (n +^
              try IntTripleMap.find (r, g, b + 1) m
              with Not_found -> 0) m in
        m) m IntTripleMap.empty) (IntTripleMap.singleton (0, 0, 0) 1) as_ in
  Printf.printf "%d\n" @@
  IntTripleMap.fold (fun _ -> ( +^ )) m 0