1815B · Sum Graph

Problem: 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).

Input Format: Each test contains multiple test cases. The first line contains a single integer $t$ ($1 \le t \le 100$) — the number of test cases.

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

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^3$.

Note: In the first test case, $n=6$ and the hidden permutation $p = [1,4,2,5,3,6]$.

Firstly, make a type $1$ query on $x=12, 2, 3$ respectively. This adds four edges to the graph in total:

• An edge that connects node $6$ and node $6$.
• An edge that connects node $1$ and node $1$.
• An edge that connects node $1$ and node $2$.
• An edge that connects node $2$ and node $1$.

Since all of these queries are valid, the interactor returns $1$ after each of them.

Then, query the number of edges in the shortest path between node $p_1 = 1$ and $p_3 = 2$, which is equal to $1$.

Then, make a type $1$ query on $x=5$. This adds four edges to the graph in total:

• An edge that connects node $1$ and node $4$.
• An edge that connects node $2$ and node $3$.
• An edge that connects node $3$ and node $2$.
• An edge that connects node $4$ and node $1$.

Since this query is valid, the interactor returns $1$.

Then, query the number of edges in the shortest path between node $p_1 = 1$ and $p_5 = 3$, which is equal to $2$.

Then, query the number of edges in the shortest path between node $p_4 = 5$ and $p_5 = 3$. Such a path doesn't exist, therefore the interactor returns $-1$.

Afterwards, due to some magic, two possible permutations that can be $p$ are determined: the first permutation is $[1,4,2,5,3,6]$ and the second permutation is $[1,2,3,4,5,6]$. Since the first permutation is equal to the hidden permutation, this test case is solved correctly. In total, $7$ queries are used, which is within the limit of $2 \cdot 6 = 12$ queries.

Since the answer is correct, the interactor returns $1$.

In the second test case, $n=2$ and the hidden permutation is $p = [2,1]$.

Since there are only $2! = 2$ possible permutations, no queries are needed. It is sufficient to just output the two permutations, $[1,2]$ and $[2,1]$. In total, $0$ queries are used, which is within the limit of $2 \cdot 2 = 4$ queries.

Since the answer is correct, the interactor returns $1$.