CF1152F2.Neko Rules the Catniverse (Large Version)

This problem is same as the previous one, but has larger constraints.

Aki is playing a new video game. In the video game, he will control Neko, the giant cat, to fly between planets in the Catniverse.

There are $n$ planets in the Catniverse, numbered from $1$ to $n$ . At the beginning of the game, Aki chooses the planet where Neko is initially located. Then Aki performs $k - 1$ moves, where in each move Neko is moved from the current planet $x$ to some other planet $y$ such that:

• Planet $y$ is not visited yet.
• $1 \leq y \leq x + m$ (where $m$ is a fixed constant given in the input)

This way, Neko will visit exactly $k$ different planets. Two ways of visiting planets are called different if there is some index $i$ such that, the $i$ -th planet visited in the first way is different from the $i$ -th planet visited in the second way.

What is the total number of ways to visit $k$ planets this way? Since the answer can be quite large, print it modulo $10^9 + 7$ .

The only line contains three integers $n$ , $k$ and $m$ ( $1 \le n \le 10^9$ , $1 \le k \le \min(n, 12)$ , $1 \le m \le 4$ ) — the number of planets in the Catniverse, the number of planets Neko needs to visit and the said constant $m$ .

Print exactly one integer — the number of different ways Neko can visit exactly $k$ planets. Since the answer can be quite large, print it modulo $10^9 + 7$ .

In the first example, there are $4$ ways Neko can visit all the planets:

• $1 \rightarrow 2 \rightarrow 3$
• $2 \rightarrow 3 \rightarrow 1$
• $3 \rightarrow 1 \rightarrow 2$
• $3 \rightarrow 2 \rightarrow 1$

In the second example, there are $9$ ways Neko can visit exactly $2$ planets:

• $1 \rightarrow 2$
• $2 \rightarrow 1$
• $2 \rightarrow 3$
• $3 \rightarrow 1$
• $3 \rightarrow 2$
• $3 \rightarrow 4$
• $4 \rightarrow 1$
• $4 \rightarrow 2$
• $4 \rightarrow 3$

In the third example, with $m = 4$ , Neko can visit all the planets in any order, so there are $5! = 120$ ways Neko can visit all the planets.

In the fourth example, Neko only visit exactly $1$ planet (which is also the planet he initially located), and there are $100$ ways to choose the starting planet for Neko.