200742: [AtCoder]ARC074 E - RGB Sequence

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