Description:
This is an interactive problem.
Little Dormi was faced with an awkward problem at the carnival: he has to guess the edges of an unweighted tree of $$$n$$$ nodes! The nodes of the tree are numbered from $$$1$$$ to $$$n$$$.
The game master only allows him to ask one type of question:
- Little Dormi picks a node $$$r$$$ ($$$1 \le r \le n$$$), and the game master will reply with an array $$$d_1, d_2, \ldots, d_n$$$, where $$$d_i$$$ is the length of the shortest path from node $$$r$$$ to $$$i$$$, for all $$$1 \le i \le n$$$.
Additionally, to make the game unfair challenge Little Dormi the game master will allow at most $$$\lceil\frac{n}{2}\rceil$$$ questions, where $$$\lceil x \rceil$$$ denotes the smallest integer greater than or equal to $$$x$$$.
Faced with the stomach-churning possibility of not being able to guess the tree, Little Dormi needs your help to devise a winning strategy!
Note that the game master creates the tree before the game starts, and does not change it during the game.
Input Format:
The first line of input contains the integer $$$n$$$ ($$$2 \le n \le 2\,000$$$), the number of nodes in the tree.
You will then begin interaction.
Output Format:
When your program has found the tree, first output a line consisting of a single "!" followed by $$$n-1$$$ lines each with two space separated integers $$$a$$$ and $$$b$$$, denoting an edge connecting nodes $$$a$$$ and $$$b$$$ ($$$1 \le a, b \le n$$$). Once you are done, terminate your program normally immediately after flushing the output stream.
You may output the edges in any order and an edge $$$(a,b)$$$ is considered the same as an edge $$$(b,a)$$$. Answering is not considered as a query.
Note:
Here is the tree from the first example.
Notice that the edges can be output in any order.
Additionally, here are the answers for querying every single node in example $$$1$$$:
- $$$1$$$: $$$[0,1,2,2]$$$
- $$$2$$$: $$$[1,0,1,1]$$$
- $$$3$$$: $$$[2,1,0,2]$$$
- $$$4$$$: $$$[2,1,2,0]$$$
Below is the tree from the second example interaction.
Lastly, here are the answers for querying every single node in example $$$2$$$:
- $$$1$$$: $$$[0,4,1,3,2]$$$
- $$$2$$$: $$$[4,0,3,1,2]$$$
- $$$3$$$: $$$[1,3,0,2,1]$$$
- $$$4$$$: $$$[3,1,2,0,1]$$$
- $$$5$$$: $$$[2,2,1,1,0]$$$