Ceil Divisions

题意翻译

给你一个长度为 $n$ 的序列 $a$ 满足 $a_i=i$。 你每次可以进行一次如下操作： - 选择两个数 $a_x,a_y$，将 $a_x$ 修改为 $\lceil\frac{a_x}{a_y}\rceil$。 你需要在 $n+5$ 步之内将这个序列 $a$ 修改为 $n-1$ 个 $1$ 和 $1$ 个 $2$ 的序列。 输出方案，即每一步操作的 $x,y$。 注意：**你不需要最小化步数**。

题目描述

You have an array $ a_1, a_2, \dots, a_n $ where $ a_i = i $ . In one step, you can choose two indices $ x $ and $ y $ ( $ x \neq y $ ) and set $ a_x = \left\lceil \frac{a_x}{a_y} \right\rceil $ (ceiling function). Your goal is to make array $ a $ consist of $ n - 1 $ ones and $ 1 $ two in no more than $ n + 5 $ steps. Note that you don't have to minimize the number of steps.

输入输出格式

输入格式

The first line contains a single integer $ t $ ( $ 1 \le t \le 1000 $ ) — the number of test cases. The first and only line of each test case contains the single integer $ n $ ( $ 3 \le n \le 2 \cdot 10^5 $ ) — the length of array $ a $ . It's guaranteed that the sum of $ n $ over test cases doesn't exceed $ 2 \cdot 10^5 $ .

输出格式

For each test case, print the sequence of operations that will make $ a $ as $ n - 1 $ ones and $ 1 $ two in the following format: firstly, print one integer $ m $ ( $ m \le n + 5 $ ) — the number of operations; next print $ m $ pairs of integers $ x $ and $ y $ ( $ 1 \le x, y \le n $ ; $ x \neq y $ ) ( $ x $ may be greater or less than $ y $ ) — the indices of the corresponding operation. It can be proven that for the given constraints it's always possible to find a correct sequence of operations.

输入输出样例

输入样例 #1

2
3
4

输出样例 #1

2
3 2
3 2
3
3 4
4 2
4 2

说明

In the first test case, you have array $ a = [1, 2, 3] $ . For example, you can do the following: 1. choose $ 3 $ , $ 2 $ : $ a_3 = \left\lceil \frac{a_3}{a_2} \right\rceil = 2 $ and array $ a = [1, 2, 2] $ ; 2. choose $ 3 $ , $ 2 $ : $ a_3 = \left\lceil \frac{2}{2} \right\rceil = 1 $ and array $ a = [1, 2, 1] $ . You've got array with $ 2 $ ones and $ 1 $ two in $ 2 $ steps.In the second test case, $ a = [1, 2, 3, 4] $ . For example, you can do the following: 1. choose $ 3 $ , $ 4 $ : $ a_3 = \left\lceil \frac{3}{4} \right\rceil = 1 $ and array $ a = [1, 2, 1, 4] $ ; 2. choose $ 4 $ , $ 2 $ : $ a_4 = \left\lceil \frac{4}{2} \right\rceil = 2 $ and array $ a = [1, 2, 1, 2] $ ; 3. choose $ 4 $ , $ 2 $ : $ a_4 = \left\lceil \frac{2}{2} \right\rceil = 1 $ and array $ a = [1, 2, 1, 1] $ .