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.

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.