E - Eternal Average

Time Limit: 2 sec / Memory Limit: 256 MB

問題文

このとき、操作ができなくなるまでこの操作を繰り返すと最終的に黒板には 1 つの有理数が書かれた状態になります。

この残った有理数の値としてありうるものの個数を 10^9 + 7 で割ったあまりを求めてください。

制約

• 1 ≦ N, M ≦ 2000
• 2 ≦ K ≦ 2000
• N+M-1K-1 で割り切れる。

入力

N M K


入力例 1

2 2 2


出力例 1

5


• 0,1 を消して \frac{1}{2} を書く。
• \frac{1}{2},1 を消して \frac{3}{4} を書く。
• 0,\frac{3}{4} を消して \frac{3}{8} を書く。

入力例 2

3 4 3


出力例 2

9


入力例 3

150 150 14


出力例 3

937426930


Score : 1600 points

Problem Statement

There are N zeros and M ones written on a blackboard. Starting from this state, we will repeat the following operation: select K of the rational numbers written on the blackboard and erase them, then write a new number on the blackboard that is equal to the arithmetic mean of those K numbers. Here, assume that N + M - 1 is divisible by K - 1.

Then, if we repeat this operation until it is no longer applicable, there will be eventually one rational number left on the blackboard.

Find the number of the different possible values taken by this rational number, modulo 10^9 + 7.

Constraints

• 1 ≦ N, M ≦ 2000
• 2 ≦ K ≦ 2000
• N + M - 1 is divisible by K - 1.

Input

The input is given from Standard Input in the following format:

N M K


Output

Print the number of the different possible values taken by the rational number that will be eventually left on the blackboard, modulo 10^9 + 7.

Sample Input 1

2 2 2


Sample Output 1

5


There are five possible values for the number that will be eventually left on the blackboard: \frac{1}{4}, \frac{3}{8}, \frac{1}{2}, \frac{5}{8} and \frac{3}{4}.

For example, \frac{3}{8} can be eventually left if we:

• Erase 0 and 1, then write \frac{1}{2}.
• Erase \frac{1}{2} and 1, then write \frac{3}{4}.
• Erase 0 and \frac{3}{4}, then write \frac{3}{8}.

Sample Input 2

3 4 3


Sample Output 2

9


Sample Input 3

150 150 14


Sample Output 3

937426930