This is an interactive problem!

salyg1n gave Alice a set $$$S$$$ of $$$n$$$ distinct integers $$$s_1, s_2, \ldots, s_n$$$ ($$$0 \leq s_i \leq 10^9$$$). Alice decided to play a game with this set against Bob. The rules of the game are as follows:

Players take turns, with Alice going first.
In one move, Alice adds one number $$$x$$$ ($$$0 \leq x \leq 10^9$$$) to the set $$$S$$$. The set $$$S$$$ must not contain the number $$$x$$$ at the time of the move.
In one move, Bob removes one number $$$y$$$ from the set $$$S$$$. The set $$$S$$$ must contain the number $$$y$$$ at the time of the move. Additionally, the number $$$y$$$ must be strictly smaller than the last number added by Alice.
The game ends when Bob cannot make a move or after $$$2 \cdot n + 1$$$ moves (in which case Alice's move will be the last one).
The result of the game is $$$\operatorname{MEX}\dagger(S)$$$ ($$$S$$$ at the end of the game).
Alice aims to maximize the result, while Bob aims to minimize it.

Let $$$R$$$ be the result when both players play optimally. In this problem, you play as Alice against the jury program playing as Bob. Your task is to implement a strategy for Alice such that the result of the game is always at least $$$R$$$.

$$$\dagger$$$ $$$\operatorname{MEX}$$$ of a set of integers $$$c_1, c_2, \ldots, c_k$$$ is defined as the smallest non-negative integer $$$x$$$ which does not occur in the set $$$c$$$. For example, $$$\operatorname{MEX}(\{0, 1, 2, 4\})$$$ $$$=$$$ $$$3$$$.

Input

The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^5$$$) - the number of test cases.

Interaction

The interaction between your program and the jury program begins with reading an integer $$$n$$$ ($$$1 \leq n \leq 10^5$$$) - the size of the set $$$S$$$ before the start of the game.

Then, read one line - $$$n$$$ distinct integers $$$s_i$$$ $$$(0 \leq s_1 \lt s_2 \lt \ldots \lt s_n \leq 10^9)$$$ - the set $$$S$$$ given to Alice.

To make a move, output an integer $$$x$$$ ($$$0 \leq x \leq 10^9$$$) - the number you want to add to the set $$$S$$$. $$$S$$$ must not contain $$$x$$$ at the time of the move. Then, read one integer $$$y$$$ $$$(-2 \leq y \leq 10^9)$$$.

If $$$0 \leq y \leq 10^9$$$ - Bob removes the number $$$y$$$ from the set $$$S$$$. It's your turn!
If $$$y$$$ $$$=$$$ $$$-1$$$ - the game is over. After this, proceed to handle the next test case or terminate the program if it was the last test case.
Otherwise, $$$y$$$ $$$=$$$ $$$-2$$$. This means that you made an invalid query. Your program should immediately terminate to receive the verdict Wrong Answer. Otherwise, it may receive any other verdict.

After printing a query 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.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$.

Do not attempt to hack this problem.

Note

In the first test case, the set $$$S$$$ changed as follows:

{$$$1, 2, 3, 5, 7$$$} $$$\to$$$ {$$$1, 2, 3, 5, 7, 8$$$} $$$\to$$$ {$$$1, 2, 3, 5, 8$$$} $$$\to$$$ {$$$1, 2, 3, 5, 8, 57$$$} $$$\to$$$ {$$$1, 2, 3, 8, 57$$$} $$$\to$$$ {$$$0, 1, 2, 3, 8, 57$$$}. In the end of the game, $$$\operatorname{MEX}(S) = 4$$$, $$$R = 4$$$.

In the second test case, the set $$$S$$$ changed as follows:

{$$$0, 1, 2$$$} $$$\to$$$ {$$$0, 1, 2, 3$$$} $$$\to$$$ {$$$1, 2, 3$$$} $$$\to$$$ {$$$0, 1, 2, 3$$$}. In the end of the game, $$$\operatorname{MEX}(S) = 4$$$, $$$R = 4$$$.

In the third test case, the set $$$S$$$ changed as follows:

{$$$5, 7, 57$$$} $$$\to$$$ {$$$0, 5, 7, 57$$$}. In the end of the game, $$$\operatorname{MEX}(S) = 1$$$, $$$R = 1$$$.

Salyg1n and the MEX Game

Examples