Span Covering

Niwango bought a piece of land that can be represented as a half-open interval [0, X).

Niwango will lay out N vinyl sheets on this land. The sheets are numbered 1,2, \ldots, N, and they are distinguishable. For Sheet i, he can choose an integer j such that 0 \leq j \leq X - L_i and cover [j, j + L_i) with this sheet.

Find the number of ways to cover the land with the sheets such that no point in [0, X) remains uncovered, modulo (10^9+7). We consider two ways to cover the land different if and only if there is an integer i (1 \leq i \leq N) such that the region covered by Sheet i is different.

Constraints

• 1 \leq N \leq 100
• 1 \leq L_i \leq X \leq 500
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

N X L_1 L_2 \ldots L_N

Output

Print the answer.

Examples

Input

3 3 1 1 2

Output

10

Input

18 477 324 31 27 227 9 21 41 29 50 34 2 362 92 11 13 17 183 119

Output

134796357

inputFormat

input are integers.

Input

Input is given from Standard Input in the following format:

N X L_1 L_2 \ldots L_N

outputFormat

Output

Print the answer.

Examples

Input

3 3 1 1 2

Output

10

Input

18 477 324 31 27 227 9 21 41 29 50 34 2 362 92 11 13 17 183 119

Output

134796357

样例

3 3
1 1 2

10