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 Rust (1.15.1) (judge verdict: Accepted)

#![allow(non_snake_case)]
#![allow(unused_variables)]
#![allow(dead_code)]

fn main() {
    let N: usize = {
        let mut line: String = String::new();
        std::io::stdin().read_line(&mut line).unwrap();
        line.trim().parse().unwrap()
    };
    let A: Vec<i64> = {
        let mut line: String = String::new();
        std::io::stdin().read_line(&mut line).unwrap();
        line.split_whitespace()
            .map(|x| x.parse().unwrap())
            .collect()
    };

    const MOD: i64 = 1000000007;
    fn modinv(x: i64) -> i64 {
        let mut a = x;
        let mut b = MOD;
        let mut u = 1;
        let mut v = 0;
        while b != 0 {
            let t = a / b;
            a -= t * b;
            std::mem::swap(&mut a, &mut b);
            u -= t * v;
            std::mem::swap(&mut u, &mut v);
        }
        u %= MOD;
        if u >= 0 {
            u
        } else {
            u + MOD
        }
    }
    #[derive(Copy, Clone)]
    struct ModInt {
        x: i64,
    }
    impl ModInt {
        fn new(x: i64) -> ModInt {
            let y = if x >= 0 {
                x % MOD
            } else {
                (x + (1 - x / MOD) * MOD) % MOD
            };
            ModInt { x: y }
        }
    }
    impl std::ops::Add<ModInt> for ModInt {
        type Output = ModInt;
        fn add(self, other: ModInt) -> ModInt {
            ModInt::new(self.x + other.x)
        }
    }
    impl std::ops::Add<i64> for ModInt {
        type Output = ModInt;
        fn add(self, other: i64) -> ModInt {
            ModInt::new(self.x + other)
        }
    }
    impl std::ops::Add<usize> for ModInt {
        type Output = ModInt;
        fn add(self, other: usize) -> ModInt {
            ModInt::new(self.x + (other as i64))
        }
    }
    impl std::ops::Sub<ModInt> for ModInt {
        type Output = ModInt;
        fn sub(self, other: ModInt) -> ModInt {
            ModInt::new(self.x - other.x)
        }
    }
    impl std::ops::Sub<i64> for ModInt {
        type Output = ModInt;
        fn sub(self, other: i64) -> ModInt {
            ModInt::new(self.x - other)
        }
    }
    impl std::ops::Sub<usize> for ModInt {
        type Output = ModInt;
        fn sub(self, other: usize) -> ModInt {
            ModInt::new(self.x - (other as i64))
        }
    }
    impl std::ops::Mul<ModInt> for ModInt {
        type Output = ModInt;
        fn mul(self, other: ModInt) -> ModInt {
            ModInt::new(self.x * other.x)
        }
    }
    impl std::ops::Mul<i64> for ModInt {
        type Output = ModInt;
        fn mul(self, other: i64) -> ModInt {
            ModInt::new(self.x * other)
        }
    }
    impl std::ops::Mul<usize> for ModInt {
        type Output = ModInt;
        fn mul(self, other: usize) -> ModInt {
            ModInt::new(self.x * (other as i64))
        }
    }
    impl std::ops::Div<ModInt> for ModInt {
        type Output = ModInt;
        fn div(self, other: ModInt) -> ModInt {
            ModInt::new(self.x * modinv(other.x as i64))
        }
    }
    impl std::ops::Div<i64> for ModInt {
        type Output = ModInt;
        fn div(self, other: i64) -> ModInt {
            ModInt::new(self.x * modinv(other))
        }
    }
    impl std::ops::Div<usize> for ModInt {
        type Output = ModInt;
        fn div(self, other: usize) -> ModInt {
            ModInt::new(self.x * modinv(other as i64))
        }
    }
    impl std::fmt::Display for ModInt {
        fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
            write!(f, "{}", self.x)
        }
    }

    let (ans, _) = A
        .iter()
        .fold((ModInt::new(1), vec![0, 0, 0]), |(res, c), &a| {
            (
                res * c.iter().filter(|&&x| x == a).count(),
                (0..3)
                    .map(|i| {
                        if c[i] == a && (0..i).all(|j| c[j] != a) {
                            c[i] + 1
                        } else {
                            c[i]
                        }
                    })
                    .collect::<Vec<_>>(),
            )
        });

    println!("{}", ans);
}