CF954F.Runner's Problem

You are running through a rectangular field. This field can be represented as a matrix with $3$ rows and $m$ columns. $(i,j)$ denotes a cell belonging to $i$ -th row and $j$ -th column.

You start in $(2,1)$ and have to end your path in $(2,m)$ . From the cell $(i,j)$ you may advance to:

• $(i-1,j+1)$ — only if i>1 ,
• $(i,j+1)$ , or
• $(i+1,j+1)$ — only if i<3 .

However, there are $n$ obstacles blocking your path. $k$ -th obstacle is denoted by three integers $a_{k}$ , $l_{k}$ and $r_{k}$ , and it forbids entering any cell $(a_{k},j)$ such that $l_{k}<=j<=r_{k}$ .

You have to calculate the number of different paths from $(2,1)$ to $(2,m)$ , and print it modulo $10^{9}+7$ .

The first line contains two integers $n$ and $m$ ( $1<=n<=10^{4}$ , $3<=m<=10^{18}$ ) — the number of obstacles and the number of columns in the matrix, respectively.

Then $n$ lines follow, each containing three integers $a_{k}$ , $l_{k}$ and $r_{k}$ ( $1<=a_{k}<=3$ , $2<=l_{k}<=r_{k}<=m-1$ ) denoting an obstacle blocking every cell $(a_{k},j)$ such that $l_{k}<=j<=r_{k}$ . Some cells may be blocked by multiple obstacles.

Print the number of different paths from $(2,1)$ to $(2,m)$ , taken modulo $10^{9}+7$ . If it is impossible to get from $(2,1)$ to $(2,m)$ , then the number of paths is $0$ .