CF1956C Nene's Magical Matrix

The magical girl Nene has an $ n\times n $ matrix $ a $ filled with zeroes. The $ j $ -th element of the $ i $ -th row of matrix $ a $ is denoted as $ a_{i, j} $ . She can perform operations of the following two types with this matrix: - Type $ 1 $ operation: choose an integer $ i $ between $ 1 $ and $ n $ and a permutation $ p_1, p_2, \ldots, p_n $ of integers from $ 1 $ to $ n $ . Assign $ a_{i, j}:=p_j $ for all $ 1 \le j \le n $ simultaneously. - Type $ 2 $ operation: choose an integer $ i $ between $ 1 $ and $ n $ and a permutation $ p_1, p_2, \ldots, p_n $ of integers from $ 1 $ to $ n $ . Assign $ a_{j, i}:=p_j $ for all $ 1 \le j \le n $ simultaneously. Nene wants to maximize the sum of all the numbers in the matrix $ \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}a_{i,j} $ . She asks you to find the way to perform the operations so that this sum is maximized. As she doesn't want to make too many operations, you should provide a solution with no more than $ 2n $ operations. A permutation of length $ n $ is an array consisting of $ n $ distinct integers from $ 1 $ to $ n $ in arbitrary order. For example, $ [2,3,1,5,4] $ is a permutation, but $ [1,2,2] $ is not a permutation ( $ 2 $ appears twice in the array), and $ [1,3,4] $ is also not a permutation ( $ n=3 $ but there is $ 4 $ in the array).

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 500 $ ). The description of test cases follows. The only line of each test case contains a single integer $ n $ ( $ 1 \le n \le 500 $ ) — the size of the matrix $ a $ . It is guaranteed that the sum of $ n^2 $ over all test cases does not exceed $ 5 \cdot 10^5 $ .

For each test case, in the first line output two integers $ s $ and $ m $ ( $ 0\leq m\leq 2n $ ) — the maximum sum of the numbers in the matrix and the number of operations in your solution. In the $ k $ -th of the next $ m $ lines output the description of the $ k $ -th operation: - an integer $ c $ ( $ c \in \{1, 2\} $ ) — the type of the $ k $ -th operation; - an integer $ i $ ( $ 1 \le i \le n $ ) — the row or the column the $ k $ -th operation is applied to; - a permutation $ p_1, p_2, \ldots, p_n $ of integers from $ 1 $ to $ n $ — the permutation used in the $ k $ -th operation. Note that you don't need to minimize the number of operations used, you only should use no more than $ 2n $ operations. It can be shown that the maximum possible sum can always be obtained in no more than $ 2n $ operations.

In the first test case, the maximum sum $ s=1 $ can be obtained in $ 1 $ operation by setting $ a_{1, 1}:=1 $ . In the second test case, the maximum sum $ s=7 $ can be obtained in $ 3 $ operations as follows: It can be shown that it is impossible to make the sum of the numbers in the matrix larger than $ 7 $ .