You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in c++17.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```cpp
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
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.

## Input Format
The 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.

## Output Format
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 $$$10^{-8}$$$.

## Examples
```input
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
```

## Note
In 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.

Now solve the problem and return the code.