Contest Duration: - (local time) (100 minutes) Back to Home
E - Distance Sequence /

Time Limit: 2 sec / Memory Limit: 1024 MB

### 問題文

• 1\le A_i \le M (1 \le i \le N)

• |A_i - A_{i+1}| \geq K (1 \le i \le N - 1)

ただし、答えは非常に大きくなることがあるので、答えを 998244353 で割った余りを求めてください。

### 制約

• 2 \leq N \leq 1000
• 1 \leq M \leq 5000
• 0 \leq K \leq M-1
• 入力は全て整数

### 入力

N M K


### 入力例 1

2 3 1


### 出力例 1

6


• (1,2)
• (1,3)
• (2,1)
• (2,3)
• (3,1)
• (3,2)

### 入力例 2

3 3 2


### 出力例 2

2


• (1,3,1)
• (3,1,3)

### 入力例 3

100 1000 500


### 出力例 3

657064711


Score : 500 points

### Problem Statement

How many integer sequences A=(A_1,\ldots,A_N) of length N satisfy all the conditions below?

• 1\le A_i \le M (1 \le i \le N)

• |A_i - A_{i+1}| \geq K (1 \le i \le N - 1)

Since the count can be enormous, find it modulo 998244353.

### Constraints

• 2 \leq N \leq 1000
• 1 \leq M \leq 5000
• 0 \leq K \leq M-1
• All values in input are integers.

### Input

Input is given from Standard Input in the following format:

N M K


### Output

Print the count modulo 998244353.

### Sample Input 1

2 3 1


### Sample Output 1

6


The following 6 sequences satisfy the conditions.

• (1,2)
• (1,3)
• (2,1)
• (2,3)
• (3,1)
• (3,2)

### Sample Input 2

3 3 2


### Sample Output 2

2


The following 2 sequences satisfy the conditions.

• (1,3,1)
• (3,1,3)

### Sample Input 3

100 1000 500


### Sample Output 3

657064711


Print the count modulo 998244353.