Ehab and the Big Finale
Description
This is an interactive problem.
You're given a tree consisting of $n$ nodes, rooted at node $1$. A tree is a connected graph with no cycles.
We chose a hidden node $x$. In order to find this node, you can ask queries of two types:
- d $u$ ($1 \le u \le n$). We will answer with the distance between nodes $u$ and $x$. The distance between two nodes is the number of edges in the shortest path between them.
- s $u$ ($1 \le u \le n$). We will answer with the second node on the path from $u$ to $x$. However, there's a plot twist. If $u$ is not an ancestor of $x$, you'll receive "Wrong answer" verdict!
Node $a$ is called an ancestor of node $b$ if $a \ne b$ and the shortest path from node $1$ to node $b$ passes through node $a$. Note that in this problem a node is not an ancestor of itself.
Can you find $x$ in no more than $36$ queries? The hidden node is fixed in each test beforehand and does not depend on your queries.
The first line contains the integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the number of nodes in the tree.
Each of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \le u,v \le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.
To print the answer, print "! x" (without quotes).
Interaction
To ask a question, print it in one of the formats above:
- d $u$ ($1 \le u \le n$), or
- s $u$ ($1 \le u \le n$).
After each question, you should read the answer: either the distance or the second vertex on the path, as mentioned in the legend.
If we answer with $-1$ instead of a valid answer, that means you exceeded the number of queries, made an invalid query, or violated the condition in the second type of queries. Exit immediately after receiving $-1$ and you will see Wrong answer verdict. Otherwise, you can get an arbitrary verdict because your solution will continue to read from a closed stream.
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 the documentation for other languages.
Hacks:
The first line should contain two integers $n$ and $x$ ($2 \le n \le 2 \cdot 10^5$, $1 \le x \le n$).
Each of the next $n-1$ lines should contain two integers $u$ and $v$ ($1 \le u,v \le n$) that mean there is an edge between nodes $u$ and $v$. The edges must form a tree.
Input
The first line contains the integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the number of nodes in the tree.
Each of the next $n-1$ lines contains two space-separated integers $u$ and $v$ ($1 \le u,v \le n$) that mean there's an edge between nodes $u$ and $v$. It's guaranteed that the given graph is a tree.
Output
To print the answer, print "! x" (without quotes).
Samples
5
1 2
1 3
3 4
3 5
3
5
d 2
s 3
! 5
Note
In the first example, the hidden node is node $5$.
We first ask about the distance between node $x$ and node $2$. The answer is $3$, so node $x$ is either $4$ or $5$. We then ask about the second node in the path from node $3$ to node $x$. Note here that node $3$ is an ancestor of node $5$. We receive node $5$ as the answer. Finally, we report that the hidden node is node $5$.