Okabe and El Psy Kongroo

你在 $(0,0)$ 。 在 $(x,y)$ 时，每次移动可以到达 $(x+1,y+1),(x+1,y),(x+1,y-1)$ 。 平面上有 $n$ 条线段，平行于 $x$ 轴，参数为$a_i$ , $b_i$ ,$c_i$ ，表示在 $(a_i,c_i)$ 到 $(b_i,c_i)$ 的一条线段，保证 $b_i=a_{i+1}$。 要求你一直在线段的下方且在 $x$ 轴上方，即 $a_i \leq x \leq b_i$ 时， $0 \leq y \leq c_i$ 。 问 $:$ 到达 $(k,0)$ 的方案数，方案数对 $10^9+7$ 取模。 $n \leq 100 , k \leq 10^{18} , c_i \leq 15$

Okabe likes to take walks but knows that spies from the Organization could be anywhere; that's why he wants to know how many different walks he can take in his city safely. Okabe's city can be represented as all points $ (x,y) $ such that $ x $ and $ y $ are non-negative. Okabe starts at the origin (point $ (0,0) $ ), and needs to reach the point $ (k,0) $ . If Okabe is currently at the point $ (x,y) $ , in one step he can go to $ (x+1,y+1) $ , $ (x+1,y) $ , or $ (x+1,y-1) $ . Additionally, there are $ n $ horizontal line segments, the $ i $ -th of which goes from $ x=a_{i} $ to $ x=b_{i} $ inclusive, and is at $ y=c_{i} $ . It is guaranteed that $ a_{1}=0 $ , $ a_{n}<=k<=b_{n} $ , and $ a_{i}=b_{i-1} $ for $ 2<=i<=n $ . The $ i $ -th line segment forces Okabe to walk with $ y $ -value in the range $ 0<=y<=c_{i} $ when his $ x $ value satisfies $ a_{i}<=x<=b_{i} $ , or else he might be spied on. This also means he is required to be under two line segments when one segment ends and another begins. Okabe now wants to know how many walks there are from the origin to the point $ (k,0) $ satisfying these conditions, modulo $ 10^{9}+7 $ .

The first line of input contains the integers $ n $ and $ k $ ( $ 1<=n<=100 $ , $ 1<=k<=10^{18} $ ) — the number of segments and the destination $ x $ coordinate. The next $ n $ lines contain three space-separated integers $ a_{i} $ , $ b_{i} $ , and $ c_{i} $ ( $ 0<=a_{i}&lt;b_{i}<=10^{18} $ , $ 0<=c_{i}<=15 $ ) — the left and right ends of a segment, and its $ y $ coordinate. It is guaranteed that $ a_{1}=0 $ , $ a_{n}<=k<=b_{n} $ , and $ a_{i}=b_{i-1} $ for $ 2<=i<=n $ .

Print the number of walks satisfying the conditions, modulo $ 1000000007 $ ( $ 10^{9}+7 $ ).

1 3
0 3 3

4

2 6
0 3 0
3 10 2

4

The graph above corresponds to sample 1. The possible walks are: -  -  -  -  The graph above corresponds to sample 2. There is only one walk for Okabe to reach $ (3,0) $ . After this, the possible walks are: -  -  -  -