K. Uniform Chemistrytime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputIn a parallel universe there are n chemical elements, numbered from 1 to n. The element number n has not been discovered so far, and its discovery would be a pinnacle of research and would bring the person who does it eternal fame and the so-called SWERC prize.There are m independent researchers, numbered from 1 to m, that are trying to discover it. Currently, the i-th researcher has a sample of the element s_i. Every year, each researcher independently does one fusion experiment. In a fusion experiment, if the researcher currently has a sample of element a, they produce a sample of an element b that is chosen uniformly at random between a+1 and n, and they lose the sample of element a. The elements discovered by different researchers or in different years are completely independent.The first researcher to discover element n will get the SWERC prize. If several researchers discover the element in the same year, they all get the prize. For each i = 1, \, 2, \, \dots, \, m, you need to compute the probability that the i-th researcher wins the prize.InputThe first line contains two integers n and m (2 \le n \le 10^{18}, 1 \le m \le 100) — the number of elements and the number of researchers.The second line contains m integers s_1, \, s_2, \, \dots, \, s_m (1 \le s_i < n) — the elements that the researchers currently have.OutputPrint m floating-point numbers. The i-th number should be the probability that the i-th researcher wins the SWERC prize. Your answer is accepted if each number differs from the correct number by at most 10^{-8}.ExamplesInput
2 3
1 1 1
Output
1.0 1.0 1.0
Input
3 3
1 1 2
Output
0.5 0.5 1.0
Input
3 3
1 1 1
Output
0.625 0.625 0.625
Input
100 7
1 2 4 8 16 32 64
Output
0.178593469 0.179810455 0.182306771
0.187565366 0.199300430 0.229356322
0.348722518
NoteIn the first sample, all researchers will discover element 2 in the first year and win the SWERC prize.In the second sample, the last researcher will definitely discover element 3 in the first year and win the SWERC prize. The first two researchers have a 50\% chance of discovering element 2 and a 50\% chance of discovering element 3, and only element 3 will bring them the prize.In the third sample, each researcher has an independent 50\% chance of discovering element 3 in the first year, in which case they definitely win the SWERC prize. Additionally, if they all discover element 2 in the first year, which is a 12.5\% chance, then they will all discover element 3 in the second year and all win the prize.
2 3
1 1 1

1.0 1.0 1.0 
['dp', 'math', '*3200']