Description

Mr. Furion is a math teacher. His students are very lazy and they do not like to do their homework. One day, Mr. Furion decides to give them a special problem in order to see whether his students are talents in math or they are just too lazy to do their homework. The problem is:

Given an integer k, n integers m₁

,m

₂

…m

_n

, and a formula below:

X₁

xor X

₂

xor X

₃

… xor X

_n

= k

Please figure out that how many integral solutions of the formula can satisfy: 0<=X_i

<=m

_i

(i=1…n)

Input

There are at most 100 test cases.

The first line of each test case contains two integers, n and k. The second line of each test case contains n integers: m₁

,m

₂

…m

_n

. The meaning of n,k, m

₁

,m

₂

…m

_n

are described above. (1<=n<=50,0<=k,m

₁

,m

₂

…m

_n

<=2

³¹

-1 )

The input is ended by “0 0”

Output

You should output an integer for each test case, which is the number of solutions. As the number might be very large, you should only output the number modulo 1000000003.

11 2047
1024 512 256 128 64 32 16 8 4 2 1
10 2047
1024 512 256 128 64 32 16 8 4 2 
0 0

1
0

Source