Codeforces Round 554 (Div. 2) |
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Finished |

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F2. Neko Rules the Catniverse (Large Version)

time limit per test

7 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputThis 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 $$$10^9 + 7$$$.

Input

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 $$$m$$$.

Output

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 $$$10^9 + 7$$$.

Examples

Input

3 3 1

Output

4

Input

4 2 1

Output

9

Input

5 5 4

Output

120

Input

100 1 2

Output

100

Note

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.

Codeforces (c) Copyright 2010-2023 Mike Mirzayanov

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