Lost Tree

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.

After taking input, you may make at most $$$\lceil\frac{n}{2}\rceil$$$ queries. Each query is made in the format "? r", where $$$r$$$ is an integer $$$1 \le r \le n$$$ that denotes the node you want to pick for that query.

You will then receive $$$n$$$ space separated integers $$$d_1, d_2, \ldots, d_n$$$, where $$$d_i$$$ is the length of the shortest path from node $$$r$$$ to $$$i$$$, followed by a newline.

After printing a query do not forget to output end of line and flush the output. Otherwise, you will get Idleness limit exceeded. To do this, use:

• fflush(stdout) or cout.flush() in C++;
• System.out.flush() in Java;
• flush(output) in Pascal;
• stdout.flush() in Python;
• see documentation for other languages.

If at any point you make an invalid query or try to make more than $$$\lceil \frac{n}{2} \rceil$$$ queries, the interaction will terminate immediately and you will receive a Wrong Answer verdict.

Hacks

To hack a solution, use the following format.

The first line contains the integer $$$n$$$ ($$$2 \le n \le 2\,000$$$).

The next $$$n−1$$$ lines contain two integers $$$u$$$ and $$$v$$$ ($$$1 \le u,v \le n$$$) denoting an edge between $$$u$$$ and $$$v$$$ ($$$u \neq v$$$). These $$$n-1$$$ edges must form a tree.