2017/round2/rainovernewyork.html

<p>
Pierre Peintre is slaving away over a new abstract painting entitled <i>Rain Over New York</i>.
This will be a simple yet powerful piece, omitting incidental details such as busy city dwellers shielding themselves with umbrellas,
and instead focusing on the fundamental atmosphere of a rainy metropolitan day.
It will be painted on a canvas which is subdivided into a grid of 1cm x 1cm cells, with <strong>N</strong> rows and <strong>M</strong> columns. 
Each cell in this grid will be filled in with a solid color, either black, white, grey, or blue.
</p>

<p>
The lower portion of <i>Rain Over New York</i> will depict the skyline of New York City. 
In each column <em>i</em>, the bottom-most <strong>H<sub>i</sub></strong> cells will be painted grey to represent an austere skyscraper.
</p>

<p>
Somewhere above the buildings, Pierre will place a single, innocent raincloud. In particular, the cloud can be any rectangle of white cells on the canvas, as long as none of them are supposed to be grey.
</p>

<p>
Below the cloud, there must be a gentle rainfall, of course. Every cell which has a white cell somewhere directly above it and a grey cell somewhere directly below it, and which isn't supposed to be white or grey itself, should be painted blue. 
Note that there may be no such cells, if the cloud is immediately above the skyline.
</p>

<p>
All of the remaining cells in the painting will be painted black, providing a serene nighttime backdrop for the scene.
</p>

<p>
Pierre knows that every painting he can produce like this will sell for an enormous sum of money, but only if it's unique. 
As such, he'll paint as many different paintings as he can by varying the position and dimensions of the raincloud depicted in them. 
Two paintings are considered distinct if at least one cell on the canvas is a different color in one painting than it is in the other.
</p>

<p>
As an example, below is an illustration of 1 of the 246 possible paintings for the fourth sample case:
</p>

<img src="{{PHOTO_ID:374610330166776}}" />

<p>
Thanks to the incredible sum of money which Pierre is sure to make from these works, he'll be able to purchase all of the paint that he'll need. 
He always buys his paint in cans of a fixed size, each of which contains just enough to cover a surface of 1,000,000,007 cm<sup>2</sup>, 
and for each color, he'll buy just enough such cans in order to be able to complete all possible distinct variations of his painting, once each. 
Always one to plan ahead, Pierre would like to figure out exactly how much paint of each color he'll have left over when he's done.
</p>

<p>
The sequence <strong>H<sub>1..M</sub></strong> can be constructed by concatenating <strong>K</strong> temporary
sequences of values <strong>S<sub>1..K</sub></strong>, the <em>i</em>th of which has a length of <strong>L<sub>i</sub></strong>.
It's guaranteed that the sum of these sequences' lengths is equal to <strong>M</strong>.
For each sequence <strong>S<sub>i</sub></strong>, you're given <strong>S<sub>i,1</sub></strong>,
and <strong>S<sub>i,2..L<sub>i</sub></sub></strong> may then be calculated as follows,
using given constants <strong>A<sub>i</sub></strong> and <strong>B<sub>i</sub></strong>:
</p>

<p>
<strong>S<sub>i,j</sub></strong> = ((<strong>A<sub>i</sub></strong> * <strong>S<sub>i,j-1</sub></strong> + <strong>B<sub>i</sub></strong>) % (<strong>N</strong> - 1)) + 1 
</p>

<h3>Input</h3>

<p>
Input begins with an integer <strong>T</strong>, the number of different base skylines Pierre wants to use.
For each skyline, there is first a line containing the three space-separated integers, <strong>N</strong>, <strong>M</strong>, and <strong>K</strong>.
Then <strong>K</strong> lines follow, the <em>i</em>th of which contains the four space-separated integers 
<strong>L<sub>i</sub></strong>, <strong>S<sub>i,1</sub></strong>, <strong>A<sub>i</sub></strong>, and <strong>B<sub>i</sub></strong>.
</p>

<h3>Output</h3>

<p>
For the <em>i</em>th skyline, print a line containing "Case #<strong>i</strong>: " followed by four space-separated integers,
the total amount of black, white, grey, and blue paint which Pierre will have left over, respectively (in cm<sup>2</sup>), after completing all possible variations of his painting.
</p>

<h3>Constraints</h3>

<p>
1 &le; <strong>T</strong> &le; 100 <br />
2 &le; <strong>N</strong> &le; 1,000,000,000 <br />
1 &le; <strong>M</strong> &le; 200,000 <br />
1 &le; <strong>K</strong> &le; 100 <br />
1 &le; <strong>L<sub>i</sub></strong> &le; M <br />
1 &le; <strong>H<sub>i</sub></strong>, <strong>S<sub>i,j</sub></strong> &le; <strong>N</strong> - 1 <br />
0 &le; <strong>A<sub>i</sub></strong>, <strong>B<sub>i</sub></strong> &lt; <strong>N</strong> - 1 <br />
</p>

<h3>Explanation of Sample</h3>

<p>
In the first case, there's only one possible painting, with the top cell painted white, and the remaining two cells painted grey. Pierre will buy 1 can each of white and grey paint, and have 1,000,000,006 and 1,000,000,005 cm<sup>2</sup> left over of those colors, respectively.
</p>

<p>
In the second case, there are 6 possible paintings: three with the cloud covering one cell, two with the cloud covering two cells, and one with the cloud covering three cells. Therefore, Pierre will use 10 cm<sup>2</sup> of white paint in total.
</p>

<p>
The fourth case corresponds to the picture shown above.
</p>