Two Different

You are given an integer $$$n$$$.

You should find a list of pairs $$$(x_1, y_1)$$$, $$$(x_2, y_2)$$$, ..., $$$(x_q, y_q)$$$ ($$$1 \leq x_i, y_i \leq n$$$) satisfying the following condition.

Let's consider some function $$$f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$$ (we define $$$\mathbb{N}$$$ as the set of positive integers). In other words, $$$f$$$ is a function that returns a positive integer for a pair of positive integers.

Let's make an array $$$a_1, a_2, \ldots, a_n$$$, where $$$a_i = i$$$ initially.

You will perform $$$q$$$ operations, in $$$i$$$-th of them you will:

1. assign $$$t = f(a_{x_i}, a_{y_i})$$$ ($$$t$$$ is a temporary variable, it is used only for the next two assignments);
2. assign $$$a_{x_i} = t$$$;
3. assign $$$a_{y_i} = t$$$.

In other words, you need to simultaneously change $$$a_{x_i}$$$ and $$$a_{y_i}$$$ to $$$f(a_{x_i}, a_{y_i})$$$. Note that during this process $$$f(p, q)$$$ is always the same for a fixed pair of $$$p$$$ and $$$q$$$.

In the end, there should be at most two different numbers in the array $$$a$$$.

It should be true for any function $$$f$$$.

Find any possible list of pairs. The number of pairs should not exceed $$$5 \cdot 10^5$$$.