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## 문제

In 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.

## 입력

The first line contains two integers $n$ and $m$ ($2 ≤ n ≤ 100$, $1 ≤ m ≤ 10$) — the number of elements and the number of researchers.

The second line contains $m$ integers $s_1, s_2, \dots , s_m$ ($1 ≤ s_i < n$) — the elements that the researchers currently have.

## 출력

Print $m$ 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 $10^{-8}$.

## 예제 입력 1

2 3
1 1 1


## 예제 출력 1

1.0 1.0 1.0


All researchers will discover element $2$ in the first year and win the SWERC prize.

## 예제 입력 2

3 3
1 1 2


## 예제 출력 2

0.5 0.5 1.0


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.

## 예제 입력 3

3 3
1 1 1


## 예제 출력 3

0.625 0.625 0.625


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.

## 예제 입력 4

100 7
1 2 4 8 16 32 64


## 예제 출력 4

0.178593469 0.179810455 0.182306771 0.187565366 0.199300430 0.229356322 0.348722518