Centroid Guess
Description
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.
Interaction
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.
Input
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.
5
2
1
2
3
1
1
1
? 1 2
? 1 3
? 1 4
? 1 5
? 2 3
? 3 4
? 4 5
! 3
Note
Here is an image of the tree from the sample.
