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#8188

자카드 유사도 1s 1024MB

Problems

Farmer John's N (1≤N≤5⋅10^5) cows are each assigned a bitstring of length K that is not all zero (1≤K≤20). Different cows may be assigned the same bitstring.

The Jaccard similarity of two bitstrings is defined as the number of set bits in their bitwise intersection divided by the number of set bits in their bitwise union. For example, the Jaccard similarity of the bitstrings 11001 and 11010 would be 2/4.

For each cow, output the sum of her bitstring's Jaccard similarity with each of the N cows' bitstrings including her own, modulo 10^9+7. Specifically, if the sum is equal to a rational number a/b where a and b are integers sharing no common factors, output the unique integer x in the range [0,109+7) such that bx−a is divisible by 10^9+7.


Input

The first line contains N and K.

The next N lines each contain an integer i∈(0,2^K) , representing a cow associated with the length-K binary representation of i.


Output

Output the sum modulo 10^9+7 for each cow on a separate line.


Example 

4 2
1
1
2
3
500000006
500000006
500000005
500000006

The cows are associated with the following bitstrings: [01,01,10,11].

For the first cow, the sum is sim(1,1)+sim(1,1)+sim(1,2)+sim(1,3)=1+1+0+1/2≡500000006(mod10^9+7).

The second cow's bitstring is the same as the first cow's, so her sum is the same as above.

For the third cow, the sum issim(2,1)+sim(2,1)+sim(2,2)+sim(2,3)=0+0+1+1/2≡500000005(mod10^9+7).


Source

USACO 2024 December Platinum

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