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 $1000000007$.

6
0 1 2 3 4 5

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

3
0 0 0

6

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

115295190