3-colorings

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 $3n$ 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 obtain the required graph with exactly $6k$ 3-colorings.

Can you do it?

Input

There is no input for this problem.

Output

First, output $n$ and $m(1\le n\le 19,1\le m\le n\cdot (n-1)/2)$ — 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 $6k$.

Examples

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