1469D · Ceil Divisions

Problem: 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.

Input Format: 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 $2 \cdot 10^5$.

Output Format: 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.

Note: 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 $a = [1, 2, 1]$.

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 $a = [1, 2, 1, 1]$.