Sum Graph

This is an interactive problem.

There is a hidden permutation $$$p_1, p_2, \dots, p_n$$$.

Consider an undirected graph with $$$n$$$ nodes only with no edges. You can make two types of queries:

1. Specify an integer $$$x$$$ satisfying $$$2 \le x \le 2n$$$. For all integers $$$i$$$ ($$$1 \le i \le n$$$) such that $$$1 \le x-i \le n$$$, an edge between node $$$i$$$ and node $$$x-i$$$ will be added.
2. Query the number of edges in the shortest path between node $$$p_i$$$ and node $$$p_j$$$. As the answer to this question you will get the number of edges in the shortest path if such a path exists, or $$$-1$$$ if there is no such path.

Note that you can make both types of queries in any order.

Within $$$2n$$$ queries (including type $$$1$$$ and type $$$2$$$), guess two possible permutations, at least one of which is $$$p_1, p_2, \dots, p_n$$$. You get accepted if at least one of the permutations is correct. You are allowed to guess the same permutation twice.

A permutation of length $$$n$$$ is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array), and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).

The interaction for each test case begins by reading the integer $$$n$$$.

Then, make at most $$$2n$$$ queries:

• If you want to make a type $$$1$$$ query, output "+ x". $$$x$$$ must be an integer between $$$2$$$ and $$$2n$$$ inclusive. After doing that read $$$1$$$ or $$$-2$$$. If you read $$$1$$$ your query was valid, otherwise it was invalid or you exceed the limit of queries, and your program must terminate immediately to receive a Wrong Answer verdict.
• If you want to make a type $$$2$$$ query, output "? i j". $$$i$$$ and $$$j$$$ must be integers between $$$1$$$ and $$$n$$$ inclusive. After that, read in a single integer $$$r$$$ ($$$-1 \le r \le n$$$) — the answer to your query. If you receive the integer $$$−2$$$ instead of an answer, it means your program has made an invalid query, or has exceeded the limit of queries. Your program must terminate immediately to receive a Wrong Answer verdict.

At any point of the interaction, if you want to guess two permutations, output "! $$$p_{1,1}$$$ $$$p_{1,2}$$$ $$$\dots$$$ $$$p_{1,n}$$$ $$$p_{2,1}$$$ $$$p_{2,2}$$$ $$$\dots$$$ $$$p_{2,n}$$$". Note that you should output the two permutations on the same line, and no exclamation mark is needed to separate the two permutations. After doing that read $$$1$$$ or $$$-2$$$. If you read $$$1$$$ your answer was correct, otherwise it was incorrect and your program must terminate immediately to receive a Wrong Answer verdict. After that, move on to the next test case, or terminate the program if there are none. Note that reporting the answer does not count as a query.

Note that even if you output a correct permutation, the second permutation should be a permutation and not an arbitrary array.

At any point, if you continue interaction after reading in the integer $$$-2$$$, you can get an arbitrary verdict because your solution will continue to read from a closed stream.

After printing a query or the answer do not forget to output the 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.

Interactor is non-adaptive. This means that all permutations are fixed before the interaction starts.

Hacks

To make a hack, use the following format.

The first line should contain a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases.

The first line of each test case should contain a single integer $$$n$$$ ($$$2 \le n \le 10^3$$$) — the length of the permutation.

The second line of each test case should contain $$$n$$$ distinct integers $$$p_1, p_2, \ldots, p_n$$$ ($$$1 \le p_i \le n$$$) — the hidden permutation.

The sum of $$$n$$$ over all test cases should not exceed $$$10^3$$$.