21907번 - 3-colorings 스페셜 저지다국어

This is an output-only problem. Note that you still have to send code which prints the output, not a text file.

A valid $3$-coloring of a graph is an assignment of colors (numbers) from the set $\{1, 2, 3\}$ to each of the $n$ vertices such that for any edge $(a, b)$ of the graph, vertices $a$ and $b$ have a different color. There are at most $3^n$ such colorings for a graph with $n$ vertices.

You work in a company, aiming to become a specialist in creating graphs with a given number of $3$-colorings. One day, you get to know that in the evening you will receive an order to produce a graph with exactly $6k$ $3$-colorings. You don’t know the exact value of $k$, only that $1 \le k \le 500$.

You don’t want to wait for the specific value of $k$ to start creating the graph. Therefore, you build a graph with at most $19$ vertices beforehand. Then, after learning that particular $k$, you are allowed to add at most $17$ edges to the graph, to obtain the required graph with exactly $6k$ $3$-colorings.

Can you do it?

There is no input for this problem.

First, output $n$ and $m$ ($1 \le n \le 19$, $1 \le m \le \frac{n(n-1)}{2}$) — the number of vertices and edges of the initial graph (the one built beforehand). Then, output $m$ lines of form $(u, v)$ — the edges of the graph.

Next, for every $k$ from $1$ to $500$ do the following:

Output $e$ — the number of edges you will add for this particular $k$ ($1 \le e \le 17$). Then, output $e$ lines of the form $(u, v)$ — the edges you will add to your graph.

There can’t be self-loops, and for every $k$, all $m+e$ edges you use have to be pairwise distinct. The number of $3$-colorings of the graph for a particular $k$ has to be exactly $6k$.

The sample output is given as an example. It contains the output for $k = 1, 2$.