| Problem Statement | |
| -------- | |
| RectJoin is the following single-player game played with square grid paper and pencil. | |
|  | |
| Let $(0, 0)$ be the coordinates of the lower left corner of the grid paper, with the $x$-axis to the right and the $y$-axis to the top. | |
| The coordinates of the upper right corner of the grid paper are $(N-1, N-1)$. | |
| Initially, $M$ dots are placed on grid points, and you can repeat the following operations as many times as possible to place dots and draw rectangles on the grid paper. | |
| In the $i$-th operation, choose a grid point $p_{i,1}$ not containing a dot and three grid points $p_{i,2}, p_{i,3}, p_{i,4}$ containing dots, which satisfy all of the following three conditions. | |
| 1. Connecting $p_{i,1} p_{i,2} p_{i,3} p_{i,4}$ in this order forms a rectangle that is parallel to the axis or inclined at 45 degrees. | |
| 2. There are no dots other than $p_{i,2}, p_{i,3}, p_{i,4}$ on the perimeter of this rectangle. | |
| 3. The perimeter of this rectangle does not share a common segment of positive length with the perimeter of an already drawn rectangle (intersecting at some points is allowed). | |
| For the chosen four points, place a new dot on $p_{i,1}$ and draw the perimeter of the rectangle $p_{i,1} p_{i,2} p_{i,3} p_{i,4}$ on the grid paper. | |
| Let $(c,c)=((N-1)/2,(N-1)/2)$ be the coordinates of the center of the graph paper. | |
| We define the weight of each grid point as $w(x,y)=(x-c)^2 + (y-c)^2 + 1$ using the distance from the center. | |
| Let $S=\sum_{x=0}^{N-1}\sum_{y=0}^{N-1} w(x,y)$ be the sum of the weights of all grid points. | |
| Let $Q$ be the set of coordinates with dots in the final state (including the initially placed dots). | |
| Then you will get the following score. | |
| \\[\mathrm{round}\left(10^6 \cdot\frac{N^2}{M}\cdot\frac{\sum_{(x, y)\in Q} w(x, y)}{S}\right)\\] | |
| Create a program to play the game to get as high a score as possible. | |
| #### Additional explanation of the rules | |
| - You must choose $p_{i,1}$ from the interior of the grid paper, i.e., from coordinates satisfying $0\leq x,y\leq N-1$. | |
| - Since a dot is placed on the grid point chosen as $p_{i,1}$, it cannot be chosen again as $p_{j,1}$ in $j(>i)$-th operation, but it can be chosen again as $p_{j,k} (k=2,3,4)$. | |
| - Similarly, $p_{i,k} (k=2,3,4)$ can be repeatedly chosen as $p_{j,k'} (k'=2,3,4)$ in $j(>i)$-th operation. | |
| - As stated in condition 2, there must be no other dots on the perimeter of the chosen rectangle, but conversely, you can choose a point on the perimeter of an already drawn rectangle as $p_{i,1}$ and place a dot on it. | |
| #### Number of test cases | |
| - Provisional test: 50 | |
| - System test: 2000. We will publish <a href="https://img.atcoder.jp/ahc014/seeds.txt">seeds.txt</a> (sha256=907b41fcba240515612a21798a10b0df7dda744b1268b74b3bbd41b93a73095e) after the contest is over. | |
| - System test contains 125 inputs for each of $N=31,33,35,\cdots,61$. | |
| - The input of seed=0 is manually created and is not included in the provisional or system test. | |
| The score of a submission is the total scores for each test case. | |
| In the provisional test, if your submission produces illegal output or exceeds the time limit for some test cases, the submission itself will be judged as <span class='label label-warning' data-toggle='tooltip' data-placement='top' title="Wrong Answer">WA</span> or <span class='label label-warning' data-toggle='tooltip' data-placement='top' title="Time Limit Exceeded">TLE</span> , and the score of the submission will be zero. | |
| The final ranking will be determined by the system test with more inputs which will be run after the contest is over. | |
| In the system test, if your submission produces illegal output or exceeds the time limit for some test cases, only the score for those test cases will be zero. | |
| The system test will be performed only for <font color="red"><strong>the last submission which received a result other than <span class="label label-warning" data-toggle="tooltip" data-placement="top" title="" data-original-title="Compilation Error">CE</span> </strong></font>. | |
| Be careful not to make a mistake in the final submission. | |
| #### About execution time | |
| Execution time may vary slightly from run to run. | |
| In addition, since system tests simultaneously perform a large number of executions, it has been observed that execution time increases by several percent compared to provisional tests. | |
| For these reasons, submissions that are very close to the time limit may result in <span class='label label-warning' data-toggle='tooltip' data-placement='top' title="Time Limit Exceeded">TLE</span> in the system test. | |
| Please measure the execution time in your program to terminate the process, or have enough margin in the execution time. | |
| Input | |
| -------- | |
| Input is given from Standard Input in the following format: | |
| ~~~ | |
| $N$ $M$ | |
| $x_1$ $y_1$ | |
| $\vdots$ | |
| $x_M$ $y_M$ | |
| ~~~ | |
| - $N$ is an odd number between 31 and 61, representing the number of vertical and horizontal grid points on the grid paper. | |
| - $M$ denotes the number of initially placed dots, satisfying $N\leq M\leq \lfloor N^2/12 \rfloor$. | |
| - $(x_1, y_1), \cdots, (x_M, y_M)$ denote the coordinates of the $M$ dots, each of which satisfies $\lfloor N/4\rfloor\leq x_i,y_i\leq\lfloor 3N/4\rfloor$. | |
| Output | |
| -------- | |
| Let $K$ be the total number of operations and $(x_{i,1}, y_{i,1}), (x_{i,2}, y_{i,2}), (x_{i,3}, y_{i,3}), (x_{i,4}, y_{i,4})$ be the four points chosen in the $i$-th operation. | |
| Then, output to Standard Output in the following format. | |
| ~~~ | |
| $K$ | |
| $x_{1,1}$ $y_{1,1}$ $x_{1,2}$ $y_{1,2}$ $x_{1,3}$ $y_{1,3}$ $x_{1,4}$ $y_{1,4}$ | |
| $\vdots$ | |
| $x_{K,1}$ $y_{K,1}$ $x_{K,2}$ $y_{K,2}$ $x_{K,3}$ $y_{K,3}$ $x_{K,4}$ $y_{K,4}$ | |
| ~~~ | |
| The order of the four points can be clockwise or counterclockwise, but $(x_{i,1}, y_{i,1})$ must be the point where the new dot is placed. | |
| <a href="https://img.atcoder.jp/ahc014/a3c240f5b1.html?lang=en&seed=0&output=20%0D%0A9+15+12+12+15+15+12+18%0D%0A15+20+12+17+15+14+18+17%0D%0A23+22+19+22+19+12+23+12%0D%0A23+14+22+15+21+14+22+13%0D%0A10+14+10+13+12+13+12+14%0D%0A11+11+12+11+12+12+11+12%0D%0A18+20+15+20+15+19+18+19%0D%0A19+16+22+19+21+20+18+17%0D%0A12+19+12+18+15+18+15+19%0D%0A15+22+12+19+15+16+18+19%0D%0A14+22+15+22+15+24+14+24%0D%0A15+8+18+11+15+14+12+11%0D%0A10+15+9+15+9+14+10+14%0D%0A11+18+12+19+10+21+9+20%0D%0A22+23+20+21+21+20+23+22%0D%0A21+15+18+15+18+14+21+14%0D%0A15+26+13+24+15+22+17+24%0D%0A20+20+16+24+14+22+18+18%0D%0A21+17+18+20+15+17+18+14%0D%0A11+14+10+13+11+12+12+13%0D%0A">Show example</a> | |
| Input Generation | |
| -------- | |
| The input of seed=0 is manually created and is not included in the provisional or system test. | |
| Let $\mathrm{rand}(L,U)$ be a function that generates a uniform random integer between $L$ and $U$, inclusive. | |
| #### Generation of $N$ and $M$ | |
| $N=\mathrm{rand}(15, 30)\times 2 + 1$, | |
| $M=\mathrm{rand}(N, \lfloor N^2/12 \rfloor)$. | |
| #### Generation of $(x_i, y_i)$ | |
| $M$ grid points $(x,y)$ satisfying $\lfloor N/4\rfloor\leq x,y\leq\lfloor 3N/4\rfloor$ are chosen at random. | |
| Tools (Input generator and visualizer) | |
| -------- | |
| - <a href="https://img.atcoder.jp/ahc014/a3c240f5b1.html?lang=en">Web version</a>: This is more powerful than the local version and can display animations and manual play. | |
| - <a href="https://img.atcoder.jp/ahc014/a3c240f5b1.zip">Local version</a>: You need a compilation environment of <a href="https://www.rust-lang.org/">Rust language</a>. | |
| - <a href="https://img.atcoder.jp/ahc014/a3c240f5b1_windows.zip">Pre-compiled binary for Windows</a>: If you are not familiar with the Rust language environment, please use this instead. | |
| <font color="red"> | |
| You are allowed to share output images (PNG) of the provided visualizer for seed=0 on twitter during the contest. | |
| Note that sharing in animation format is prohibited. | |
| </font> | |
| You have to use the specified hashtag and public account. | |
| You can only share visualization results and scores for seed=0. | |
| Do not share GIFs, output itself, scores for other seeds or mention solutions or discussions. | |
| <a href="https://twitter.com/search?q=%23AHC014%20%23visualizer&src=typed_query&f=live">List of shared images</a> | |
| {sample example} |