CF1469D Ceil Divisions

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.

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 $ .

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.

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] $ .