CF1934E.Weird LCM Operations

Given an integer $n$, you construct an array $a$ of $n$ integers, where $a_i = i$ for all integers $i$ in the range $[1, n]$. An operation on this array is defined as follows:

• Select three distinct indices $i$, $j$, and $k$ from the array, and let $x = a_i$, $y = a_j$, and $z = a_k$.
• Update the array as follows: $a_i = \operatorname{lcm}(y, z)$, $a_j = \operatorname{lcm}(x, z)$, and $a_k = \operatorname{lcm}(x, y)$, where $\operatorname{lcm}$ represents the least common multiple.

Your task is to provide a possible sequence of operations, containing at most $\lfloor \frac{n}{6} \rfloor + 5$ operations such that after executing these operations, if you create a set containing the greatest common divisors (GCDs) of all subsequences with a size greater than $1$, then all numbers from $1$ to $n$ should be present in this set.

After all the operations $a_i \le 10^{18}$ should hold for all $1 \le i \le n$.

We can show that an answer always exists.

给定一个整数 $n$，你构造一个长度为 $n$ 的整数数组 $a$，其中对所有 $i \in [1, n]$ 均有 $a_i = i$。对该数组定义如下操作：

• 从数组中选出三个互不相同的下标 $i$、$j$ 和 $k$，并令 $x = a_i$、$y = a_j$、$z = a_k$；
• 按如下方式更新数组：$a_i = \operatorname{lcm}(y, z)$，$a_j = \operatorname{lcm}(x, z)$，$a_k = \operatorname{lcm}(x, y)$，其中 $\operatorname{lcm}$ 表示最小公倍数。

你的任务是给出一个可行的操作序列，该序列至多包含 $\lfloor \frac{n}{6} \rfloor + 5$ 次操作，使得执行完这些操作后，若构造一个集合，其元素为所有长度大于 $1$ 的子序列的最大公约数（GCD），则该集合应包含 $1$ 到 $n$ 中的所有整数。

此外，所有操作完成后，需满足对所有 $1 \le i \le n$ 均有 $a_i \le 10^{18}$。

我们可以证明这样的解总是存在的。

The first line contains one integer $t$ ($1 \le t \le 10^2$) — the number of test cases. The description of the test cases follows.

The first and only line of each test case contains an integer $n$ ($3 \leq n \leq 3 \cdot 10^{4}$) — the length of the array.

It is guaranteed that the sum of $n$ over all test cases does not exceed $3 \cdot 10^{4}$.

第一行包含一个整数 $t$（$1 \le t \le 10^2$）——测试用例的数量。随后是各测试用例的描述。

每个测试用例仅有一行，包含一个整数 $n$（$3 \leq n \leq 3 \cdot 10^{4}$）——数组的长度。

保证所有测试用例的 $n$ 之和不超过 $3 \cdot 10^{4}$。

The first line should contain an integer $k$ ($0 \leq k \leq \lfloor \frac{n}{6} \rfloor + 5$) — where $k$ is the number of operations.

The next $k$ lines should contain the description of each operation i.e. $3$ integers $i$, $j$ and $k$, where $1 \leq i, j, k \leq n$ and all must be distinct.

第一行应包含一个整数 $k$（$0 \leq k \leq \lfloor \frac{n}{6} \rfloor + 5$），其中 $k$ 表示操作的次数。

接下来的 $k$ 行应描述每次操作，即每行包含三个整数 $i$、$j$ 和 $k$，满足 $1 \leq i, j, k \leq n$，且三者互不相同。

In the third test case, $a = [1, 2, 3, 4, 5, 6, 7]$.

First operation:

$i = 3$, $j = 5$, $k = 7$

$x = 3$, $y = 5$, $z = 7$.

$a = [1, 2, \operatorname{lcm}(y,z), 4, \operatorname{lcm}(x,z), 6, \operatorname{lcm}(x,y)]$ = $[1, 2, \color{red}{35}, 4, \color{red}{21}, 6, \color{red}{15}]$.

Second operation:

$i = 5$, $j = 6$, $k = 7$

$x = 21$, $y = 6$, $z = 15$.

$a = [1, 2, 35, 4, \operatorname{lcm}(y,z), \operatorname{lcm}(x,z), \operatorname{lcm}(x,y)]$ = $[1, 2, 35, 4, \color{red}{30}, \color{red}{105}, \color{red}{42}]$.

Third operation:

$i = 2$, $j = 3$, $k = 4$

$x = 2$, $y = 35$, $z = 4$.

$a = [1, \operatorname{lcm}(y,z), \operatorname{lcm}(x,z), \operatorname{lcm}(x,y), 30, 105, 42]$ = $[1, \color{red}{140}, \color{red}{4}, \color{red}{70}, 30, 105, 42]$.

Subsequences whose GCD equal to $i$ is as follows:

$\gcd(a_1, a_2) = \gcd(1, 140) = 1$

$\gcd(a_3, a_4) = \gcd(4, 70) = 2$

$\gcd(a_5, a_6, a_7) = \gcd(30, 105, 42) = 3$

$\gcd(a_2, a_3) = \gcd(140, 4) = 4$

$\gcd(a_2, a_4, a_5, a_6) = \gcd(140, 70, 30, 105) = 5$

$\gcd(a_5, a_7) = \gcd(30, 42) = 6$

$\gcd(a_2, a_4, a_6, a_7) = \gcd(140, 70, 105, 42) = 7$

在第三个测试用例中，$a = [1, 2, 3, 4, 5, 6, 7]$。

第一次操作：

$i = 3$, $j = 5$, $k = 7$

$x = 3$, $y = 5$, $z = 7$。

$a = [1, 2, \operatorname{lcm}(y,z), 4, \operatorname{lcm}(x,z), 6, \operatorname{lcm}(x,y)]$ = $[1, 2, \color{red}{35}, 4, \color{red}{21}, 6, \color{red}{15}]$。

第二次操作：

$i = 5$, $j = 6$, $k = 7$

$x = 21$, $y = 6$, $z = 15$。

$a = [1, 2, 35, 4, \operatorname{lcm}(y,z), \operatorname{lcm}(x,z), \operatorname{lcm}(x,y)]$ = $[1, 2, 35, 4, \color{red}{30}, \color{red}{105}, \color{red}{42}]$。

第三次操作：

$i = 2$, $j = 3$, $k = 4$

$x = 2$, $y = 35$, $z = 4$。

$a = [1, \operatorname{lcm}(y,z), \operatorname{lcm}(x,z), \operatorname{lcm}(x,y), 30, 105, 42]$ = $[1, \color{red}{140}, \color{red}{4}, \color{red}{70}, 30, 105, 42]$。

满足 $\gcd$ 等于 $i$ 的子序列如下：

$\gcd(a_1, a_2) = \gcd(1, 140) = 1$

$\gcd(a_3, a_4) = \gcd(4, 70) = 2$

$\gcd(a_5, a_6, a_7) = \gcd(30, 105, 42) = 3$

$\gcd(a_2, a_3) = \gcd(140, 4) = 4$

$\gcd(a_2, a_4, a_5, a_6) = \gcd(140, 70, 30, 105) = 5$

$\gcd(a_5, a_7) = \gcd(30, 42) = 6$

$\gcd(a_2, a_4, a_6, a_7) = \gcd(140, 70, 105, 42) = 7$