CF1438F.Olha and Igor

This is an interactive problem.

Igor wants to find the key to Olha's heart. The problem is, that it's at the root of a binary tree.

There is a perfect binary tree of height $h$ consisting of $n = 2^{h} - 1$ nodes. The nodes have been assigned distinct labels from $1$ to $n$ . However, Igor only knows $h$ and does not know which label corresponds to which node.

To find key to Olha's heart he needs to find the label assigned to the root by making queries of the following type at most $n+420$ times:

• Select three distinct labels $u$ , $v$ and $w$ ( $1 \leq u,v,w \leq n$ ).
• In response, Olha (the grader) will tell him the label of the lowest common ancestor of nodes labelled $u$ and $v$ , if the tree was rooted at the node labelled $w$ instead.

Help Igor to find the root!

Note: the grader is not adaptive: the labels are fixed before any queries are made.

The first and only line contains a single integer $h$ ( $3 \le h \le 18$ ) — the height of the tree.

You begin the interaction by reading $h$ .

To make a query for labels $u, v, w$ , in a separate line output "? u v w".

Numbers in the query have to satisfy $1 \le u, v, w \le n$ . Additionally, $u \ne v$ , $u \ne w$ , and $v \ne w$ .

In response, you will receive $1 \le x \le n$ , the label of the lowest common ancestor of $u$ and $v$ , if the tree was rooted at $w$ .

In case your query is invalid or you asked more than $n+420$ queries, program will print $-1$ and will finish interaction. You will receive Wrong answer verdict. Make sure to exit immediately to avoid getting other verdicts.

When you determine the label assigned to the root, output "! r", where $r$ is the label of the root.

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.

Hack Format

To hack, use the following format.

The first line should contain a single integer $h$ (height of the binary tree).

On the next line, output a permutation $p$ of size $n = 2^h - 1$ . This represents a binary tree where the root is labelled $p_1$ and for $1 < i \le n$ , the parent of $p_i$ is $p_{ \lfloor{\frac{i}{2}}\rfloor }$ .

The labels corresponding to the tree in the example are [ $4$ , $7$ , $2$ , $6$ , $1$ , $5$ , $3$ ], meaning the root is labelled $4$ , and for $1 < i \le n$ , the parent of $p_i$ is $p_{ \lfloor{\frac{i}{2}}\rfloor }$ .