You are given a polynomial \(p(x) = a_0 + a_1 x + a_2 x^2 + ... + a_m x^m\) of degree \(m\). In addition, you will have \(n\) real numbers \(x_1,...,x_n\). You need to calculate \(p(x_i)\) for each \(i \in \{1,..n\}\).

However, evaluating a polynomial gets more complex as its degree increases. Therefore you must use an approximation whenever it is possible.

To approximate \(p(x)\), you will compute \(T(x) = p(x_0) + p'(x_0) \cdot (x-x_0) + \frac{p''(x_0)}{2} \cdot (x-x_0)^2\), where \(|x - x_0| < \epsilon = 10^{-3}\). \(x_0\) can be any of the previous queries, for which you already directly computed the polynomial.

#### Input

The first line contains two integers, \(m\) and \(n\).

The next line contains \(m+1\) real numbers, which are polynomial coefficients. The first one is \(a_0\) and the last one is \(a_m\).

The last line contains \(n\) real numbers which correspond to query points where \(p(x)\) will be evaluated.

- \(0 \le m \le 5 \cdot 10^4\)
- \(1 \le n \le 5 \cdot 10^4\)
- \(a_m \neq 0\)
- \(|a_i| \leq 100\)
- \(|x_i| < 0.5\)
- \(a_i\) and \(x_i\) will not be more precise than \(10^{-4}\)

#### Output

Print the answer of each query with spaces in-between on the same single line and in the given order. Your answer will be considered correct if its absolute or relative error does not exceed \(10^{-6}\). Formally, let your answer be \(a\) and the jury's answer \(b\). Your answer will be considered correct if \(\frac{|a - b|}{max(1, |b|)} \le 10^{-6}\).

#### Example

Input 1:

```
2 3
1 0.5 3
0 0.1 0.0001
```

Output 1:

`1 1.08 1.00005`

Input 2:

```
2 3
1 1 1
0.1 0.1001 0.0999
```

Output 2:

`1.11 1.11012 1.10988`

Input 3:

```
2 3
1 1 1
0.1 0.1005 0.101
```

Output 3:

`1.11 1.1106 1.111201`

#### Explanation

**Input 1:** \(p(x) = 1 + 0.5x + 3x^2\)

First two queries \(p(0)\) and \(p(0.1)\) are computed directly. We can approximate \(p(0.0001) \approx T(0.0001) = p(0) + p'(0) \cdot (0.0001-0) + \frac{p''(0)}{2} \cdot (0.0001-0)^2\).

**Input 2:** The first query is directly calculated. The second and third ones are approximated.

**Input 3:** The first query is directly calculated. The second one is approximated. The third one is directly calculated.