Description:
This is an interactive problem.
Kostyanych has chosen a complete undirected graph$$$^{\dagger}$$$ with $$$n$$$ vertices, and then removed exactly $$$(n - 2)$$$ edges from it. You can ask queries of the following type:
- "? $$$d$$$" — Kostyanych tells you the number of vertex $$$v$$$ with a degree at least $$$d$$$. Among all possible such vertices, he selects the vertex with the minimum degree, and if there are several such vertices, he selects the one with the minimum number. He also tells you the number of another vertex in the graph, with which $$$v$$$ is not connected by an edge (if none is found, then $$$0$$$ is reported). Among all possible such vertices, he selects the one with the minimum number. Then he removes the vertex $$$v$$$ and all edges coming out of it. If the required vertex $$$v$$$ is not found, then "$$$0\ 0$$$" is reported.
Find a Hamiltonian path$$$^{\ddagger}$$$ in the original graph in at most $$$n$$$ queries. It can be proven that under these constraints, a Hamiltonian path always exists.
$$$^{\dagger}$$$A complete undirected graph is a graph in which there is exactly one undirected edge between any pair of distinct vertices. Thus, a complete undirected graph with $$$n$$$ vertices contains $$$\frac{n(n-1)}{2}$$$ edges.
$$$^{\ddagger}$$$A Hamiltonian path in a graph is a path that passes through each vertex exactly once.
Input Format:
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of test cases. The description of the test cases follows.
The only line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 10^5$$$) — the number of vertices in the graph.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$.
Output Format:
None
Note:
In the first test case, the original graph looks as follows:
Consider the queries:
- There are no vertices with a degree of at least $$$3$$$ in the graph, so "$$$0\ 0$$$" is reported.
- There are four vertices with a degree of at least $$$2$$$, and all of them have a degree of exactly $$$2$$$: $$$1$$$, $$$2$$$, $$$3$$$, $$$4$$$. Vertex $$$1$$$ is reported, because it has the minimum number, and vertex $$$4$$$ is reported, because it is the only one not connected to vertex $$$1$$$. After this, vertex $$$1$$$ is removed from the graph.
- There are three vertices with a degree of at least $$$1$$$, among them vertices $$$2$$$ and $$$3$$$ have a minimum degree of $$$1$$$ (vertex $$$4$$$ has a degree of $$$2$$$). Vertex $$$2$$$ is reported, because it has the minimum number, and vertex $$$3$$$ is reported, because it is the only one not connected to vertex $$$2$$$. After this, vertex $$$2$$$ is removed from the graph.
The path $$$4 - 3 - 1 - 2$$$ is a Hamiltonian path.
In the second test case, the original graph looks as follows:
Consider the queries:
- Vertex $$$1$$$ has a degree of at least $$$3$$$, but it is connected to all vertices, so "$$$1\ 0$$$" is reported. After this, vertex $$$1$$$ is removed from the graph.
- The remaining vertices $$$2$$$, $$$3$$$, and $$$4$$$ have a degree of at least $$$0$$$, but among them vertex $$$4$$$ has the minimum degree of $$$0$$$ (vertices $$$2$$$ and $$$3$$$ have a degree of $$$1$$$). Vertex $$$4$$$ is not connected to both vertices $$$2$$$ and $$$3$$$, so vertex $$$2$$$ is reported (as it has the minimum number). After this, vertex $$$4$$$ is removed from the graph.
The path $$$4 - 1 - 2 - 3$$$ is a Hamiltonian path.
In the third test case, the graph consists of $$$2$$$ vertices connected by an edge.