Solar cycle predictions are used by various agencies and many industry groups. The solar cycle is important for determining the lifetime of satellites in low-Earth orbit, as the drag on the satellites correlates with the solar cycle [...]. (NOAA)

Sunspot Number Progression : Observed data through May 2008 ; Dec 2012 ; Nov 2014 ; Jun 2016

The goal of the problem is to propose a perfect prediction center, with not so weak constraints.

Let us consider periodic functions from **Z** to **R**.

def f(x): return [4, -6, 7][x%3] # (with Python notations) # 4, -6, 7, 4, -6, 7, 4, -6, 7, 4, -6, 7, 4, -6, 7, ...

For example, *f* is a 3-periodic function, with *f*(0) = *f*(3) = *f*(6) = *f*(9) = 4.

With a simplified notation we will denote f as [4, -6, 7].

For two periodic functions (with integral period), the quotient of periods will be rational, in that case it can be shown that the sum of the functions is also a periodic function. Thus, the set of all such functions is a vector space over **R**.

For that problem, we consider a function that is the sum of several periodic functions all with as period an integer *N* at maximum. You will be given some starting values, you'll have to find new ones.

On the first line, you will be given an integer *N*.

On the second line, you will be given integers *y* : the first (0-indexed) *N×N* values of a periodic function *f*
that is sum of periodic functions all with as period an integer *N* at maximum.

On the third line, you will be given *N×N* integers *x*.

Print *f(x)* for all required *x*. See sample for details.

3 15 3 17 2 16 4 15 3 17 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000

16 16 16 16 16 16 16 16 16

For example *f* can be seen as the sum of three periodic functions : [10] + [5, -8] + [0, 1, 2] (with simplified notations ; periods are 1,2 and 3)

In that case *f*(10) = [10][10%1] + [5, -8][10%2] + [0, 1, 2][10%3] = 10 + 5 + 1 = 16, and so on.

N < 258 abs(y) < 10^{9}0 <= x < 10^{9}

You can safely assume output fit in a signed 32bit container.
There's 6 input files, with increasing value of *N*.
Some details (#i, *N*) :

- (#0, around 50)
- (#1, around 50)
- (#2, around 100)
- (#3, around 150)
- (#4, around 200)
- (#5, around 250)

- 3s
- 256MB

**Problem source:** Sphere Online Judge