A15501. Centroid Guess

This in an interactive problem.

There is an unknown tree consisting of $n$ nodes, which has exactly one centroid. You only know $n$ at first, and your task is to find the centroid of the tree.

You can ask the distance between any two vertices for at most $2\cdot10^5$ times.

Note that the interactor is not adaptive. That is, the tree is fixed in each test beforehand and does not depend on your queries.

A vertex is called a centroid if its removal splits the tree into subtrees with at most $\lfloor\frac{n}{2}\rfloor$ vertices each.

The only line of the input contains an integer $n$ ( $3\le n\le 7.5\cdot10^4$ ) — the number of nodes in the tree.

Start interaction by reading $n$ .

To ask a query about the distance between two nodes $u, v$ ( $1 \le u, v \le n$ ), output "? u v".

If you determine that the centroid of the tree is $x$ , use "! x" to report.

After printing a query, do not forget to output the end of a 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.

Hacks are disabled in this problem.

It's guaranteed that there are at most $500$ tests in this problem.

Here is an image of the tree from the sample.