Lost Array
Contest: Codeforces LATOKEN Round 1 (Div. 1 + Div. 2)
Difficulty rating: 2300
Tags: graphs, greedy, interactive, shortest paths

This is an interactive problem.

Note: the XOR-sum of an array $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 10^9$$$) is defined as $$$a_1 \oplus a_2 \oplus \ldots \oplus a_n$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.

Little Dormi received an array of $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ for Christmas. However, while playing with it over the winter break, he accidentally dropped it into his XOR machine, and the array got lost.

The XOR machine is currently configured with a query size of $$$k$$$ (which you cannot change), and allows you to perform the following type of query: by giving the machine $$$k$$$ distinct indices $$$x_1, x_2, \ldots, x_k$$$, it will output $$$a_{x_1} \oplus a_{x_2} \oplus \ldots \oplus a_{x_k}$$$.

As Little Dormi's older brother, you would like to help him recover the XOR-sum of his array $$$a_1, a_2, \ldots, a_n$$$ by querying the XOR machine.

Little Dormi isn't very patient, so to be as fast as possible, you must query the XOR machine the minimum number of times to find the XOR-sum of his array. Formally, let $$$d$$$ be the minimum number of queries needed to find the XOR-sum of any array of length $$$n$$$ with a query size of $$$k$$$. Your program will be accepted if you find the correct XOR-sum in at most $$$d$$$ queries.

Lastly, you also noticed that with certain configurations of the machine $$$k$$$ and values of $$$n$$$, it may not be possible to recover the XOR-sum of Little Dormi's lost array. If that is the case, you should report it as well.

The array $$$a_1, a_2, \ldots, a_n$$$ is fixed before you start querying the XOR machine and does not change with the queries.

Input format:
The only line of input contains the integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 500$$$, $$$1 \le k \le n$$$), the length of the lost array and the configured query size of the XOR machine.

Elements of the original array satisfy $$$1 \le a_i \le 10^9$$$.

It can be proven that that if it is possible to recover the XOR sum under the given constraints, it can be done in at most $$$500$$$ queries. That is, $$$d \le 500$$$.

After taking $$$n$$$ and $$$k$$$, begin interaction.

Output format:
If it is impossible to recover the XOR-sum of the array, output $$$-1$$$ immediately after taking $$$n$$$ and $$$k$$$. Do not begin interaction.

Otherwise, when your program finds the XOR-sum of the lost array $$$a_1, a_2, \ldots, a_n$$$, report the answer in the following format: "! x", where $$$x$$$ is the XOR sum of the array $$$a_1, a_2, \ldots, a_n$$$, and terminate your program normally immediately after flushing the output stream.

Note that answering does not count as a query.

Interaction format:
Each query is made in the format "? b", where $$$b$$$ is an array of exactly $$$k$$$ distinct integers from $$$1$$$ to $$$n$$$ denoting the indices of the elements in the lost array that you want to query the XOR sum of.

You will then receive an integer $$$x$$$, the XOR sum of the queried elements. It can be proven that $$$0 \le x \le 2 \cdot 10^9$$$ will always be true.

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.

If at any point you make an invalid query or try to make more than $$$500$$$ queries (which is the hard limit), the interaction will terminate immediately and give you a Wrong Answer verdict. Note that if you exceed $$$d$$$ queries, the interaction will continue normally unless you also exceed the $$$500$$$ query hard limit, though you will still receive a Wrong Answer verdict either way.

Hacks

To hack a solution, use the following format.

The first line contains the integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 500$$$, $$$1 \le k \le n$$$).

The second line contains the the array $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 10^9$$$).

Constraints:
Time limit: 1.5 s
Memory limit: 256.0 MB

Note:
In the first example interaction, the array $$$a_1, a_2, \ldots, a_n$$$ is $$$2, 1, 7, 5, 6$$$ and its XOR-sum is $$$7$$$.

The first query made asks for indices $$$1,2,3$$$, so the response is $$$a_1 \oplus a_2 \oplus a_3 = 2 \oplus 1 \oplus 7 = 4$$$.

The second query made asks for indices $$$2,3,5$$$, so the response is $$$a_2 \oplus a_3 \oplus a_5 = 1 \oplus 7 \oplus 6 = 0$$$.

The third query made asks for indices $$$4,1,5$$$, so the response is $$$a_4 \oplus a_1 \oplus a_5 = 5 \oplus 2 \oplus 6 = 1$$$. Note that the indices may be output in any order.

Additionally, even though three queries were made in the example interaction, it is just meant to demonstrate the interaction format and does not necessarily represent an optimal strategy.

In the second example interaction, there is no way to recover the XOR-sum of Little Dormi's array no matter what is queried, so the program immediately outputs $$$-1$$$ and exits.

Example input:
5 3

4

0

1

Example output:
? 1 2 3

? 2 3 5

? 4 1 5

! 7

Example input:
3 2

Example output:
-1

Editorial:
"tl;dr it's pure BFS"
The only information we can obtain from a query is the XOR sum of the subset we queried. We can also try and obtain some frequency information about the bits (whether the number of bits at position $$$i$$$ is odd for every bit $$$i$$$), but trying to combine frequency information is ultimately analogous to just combining the XOR sums of different subsets. Additionally, trying to combine XOR sums with other operations such as OR and AND will cause us to lose the information about the XOR sum of the combined subsets, and are thus not useful.
Thus, the only way to find the answer is to repeated query different subsets until their XOR sum includes every element in the array.
To find the minimum number of queries, we can run a BFS to find the shortest number of queries to go from the XOR sum of $$$0$$$ numbers to the XOR sum of all of them. Observe that only the number of values currently present in the XOR sum matters, as we can query the XOR sum of any subset of numbers in the array as long as it has size $$$k$$$. In our graph, we can transition from the XOR sum of a subset of size $$$i$$$ by picking $$$j$$$ ($$$0 \le j \le k$$$) selected numbers and $$$k-j$$$ unselected numbers. This changes the size of our subset by $$$k-2j$$$.
Time Complexity: $$$\mathcal{O}(nk)$$$