1534D · Lost Tree

Problem: 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 $2$:

• $1$: $[0,4,1,3,2]$
• $2$: $[4,0,3,1,2]$
• $3$: $[1,3,0,2,1]$
• $4$: $[3,1,2,0,1]$
• $5$: $[2,2,1,1,0]$