Problem:
Score : 800 points

Problem Statement

There are N squares arranged in a row.
The squares are numbered 1, 2, ..., N, from left to right.

Snuke is painting each square in red, green or blue.
According to his aesthetic sense, the following M conditions must all be satisfied.
The i-th condition is:

There are exactly x_i different colors among squares l_i, l_i + 1, ..., r_i.

In how many ways can the squares be painted to satisfy all the conditions?
Find the count modulo 10^9+7.

Constraints

1 ≤ N ≤ 300

1 ≤ M ≤ 300

1 ≤ l_i ≤ r_i ≤ N

1 ≤ x_i ≤ 3

Input

Input is given from Standard Input in the following format:

N M
l_1 r_1 x_1
l_2 r_2 x_2
:
l_M r_M x_M

Output

Print the number of ways to paint the squares to satisfy all the conditions, modulo 10^9+7.

Sample Input 1

3 1
1 3 3

Sample Output 1

6

The six ways are:

RGB

RBG

GRB

GBR

BRG

BGR

where R, G and B correspond to red, green and blue squares, respectively.

Sample Input 2

4 2
1 3 1
2 4 2

Sample Output 2

6

The six ways are:

RRRG

RRRB

GGGR

GGGB

BBBR

BBBG

Sample Input 3

1 3
1 1 1
1 1 2
1 1 3

Sample Output 3

0

There are zero ways.

Sample Input 4

8 10
2 6 2
5 5 1
3 5 2
4 7 3
4 4 1
2 3 1
7 7 1
1 5 2
1 7 3
3 4 2

Sample Output 4

108

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::cmp;

const MOD: usize = 1e9 as usize + 7;

fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();
    let m = sc.read();

    let mut left_x = vec![vec![]; n + 1];
    for _ in 0..m {
        let left = sc.read::<usize>();
        let right = sc.read::<usize>();
        let x: usize = sc.read();
        left_x[right].push((left, x));
    }

    let mut dp = vec![vec![vec![0; n + 1]; n + 1]; n + 1];
    dp[0][0][0] = 1;
    for r in 0..n {
        for g in 0..n {
            for b in 0..n {
                let max = cmp::max(cmp::max(r, g), b);
                let check = |a: usize, b: usize| left_x[max + 1].iter().all(|&(left, x)| {
                    let mut count = 1;
                    if a >= left { count += 1; }
                    if b >= left { count += 1; }
                    count == x
                });

                if check(g, b) {
                    dp[max + 1][g][b] += dp[r][g][b];
                    dp[max + 1][g][b] %= MOD;
                }
                if check(r, g) {
                    dp[r][g][max + 1] += dp[r][g][b];
                    dp[r][g][max + 1] %= MOD;
                }
                if check(r, b) {
                    dp[r][max + 1][b] += dp[r][g][b];
                    dp[r][max + 1][b] %= MOD;
                }
            }
        }
    }

    let mut ans = 0;
    for i in 0..n {
        for j in 0..n {
            ans += dp[i][j][n];
            ans += dp[i][n][j];
            ans += dp[n][j][i];
            ans %= MOD;
        }
    }
    println!("{}", ans);
}

pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n')
            .take_while(|&b| b != b' ' && b != b'\n')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}