Codeforces Round 939 (Div. 2) |
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Finished |

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constructive algorithms

greedy

math

*1600

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C. Nene's Magical Matrix

time limit per test

2 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputThe 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).

Input

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 $$$5 \cdot 10^5$$$.

Output

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.

Example

Input

212

Output

1 1 1 1 1 7 3 1 1 1 2 1 2 1 2 2 1 1 2

Note

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 $$$7$$$.

Codeforces (c) Copyright 2010-2024 Mike Mirzayanov

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